Equations notes tam math lab pkt

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VML COLLEGE ALGEBRA INTERMEDIATE ALGEBRA BEGINNING ALGEBRA GRE MATH THEA/ACCUPLACER

Intermediate Algebra

Tutorial 12: Graphing Equations

WTAMU > Virtual Math Lab > Intermediate Algebra

Learning Objectives

After completing this tutorial, you should be able to:

Plot points on a rectangular coordinate system.1.

Identify what quadrant or axis a point lies on.2.

Know if an equation is a linear equation.3.

Tell if an ordered pair is a solution of an equation in two variables or not.4.

Graph an equation by plotting points. 5.

Introduction

This section covers the basic ideas of graphing: rectangular coordinate system, solutions to equations in two variables,

and sketching a graph. Graphs are important in giving a visual representation of the correlation between two variables.

Even though in this section we are going to look at it generically, using a general x and y variable, you can usetwo-dimensional graphs for any application where you have two variables. For example, you may have a cost function that is

dependent on the quantity of items made. If you needed to show your boss visually the correlation of the quantity with the

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cost, you could do that on a two-dimensional graph. I believe that it is important for you learn how to do something in

general, then when you need to apply it to something specific you have the knowledge to do so. Going from general to

specific is a lot easier than specific to general. And that is what we are doing here looking at graphing in general so later you

can apply it to something specific, if needed.

Tutorial

Rectangular Coordinate System

The following is the rectangular coordinate system:

It is made up of two number lines:

The horizontal number line is the x- axis.1.

The vertical number line is the y- axis.2.

The origin is where the two intersect. This is where both number lines are 0.

It is split into four quadrants which are marked on this graph with Roman numerals.

Each point on the graph is associated with an ordered pair. When dealing with an x, y graph, x is always first

and y is always second in the ordered pair (x, y). It is a solution to an equation in two variables. Even thoughthere are two values in the ordered pair, be careful that it associates to ONLY ONE point on the graph, the

point lines up with both the x value of the ordered pair (x-axis) and the y value of the ordered pair

(y-axis).

Example 1: Plot the ordered pairs and name the quadrant or axis in which the point lies. A(2, 3), B(-1, 2),C(-3, -4), D(2, 0), and E(0, 5).

Remember that each ordered pair associates with only one point on the graph. Just line up the x value and then

the y value to get your location.

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A(2, 3) lies in quadrant I.

B(-1, 2) lies in quadrant II.

C(-3, -4) lies in quadrant III.

D(2, 0) lies on the x-axis.

E(0, 5) lies on the y-axis.

Solutions of Equations

in Two Variables

The solutions to equations in two variables consist of two values that when substituted into their corresponding variables in

the equation, make a true statement.

In other words, if your equation has two variables x and y, and you plug in a value for x and its corresponding value for y and

the mathematical statement comes out to be true, then the x and y value that you plugged in would together be a solution tothe equation.

Equations in two variables can have more than one solution.

We usually write the solutions to equations in two variables in ordered pairs.

Example 2: Determine whether each ordered pair is a solution of the given equation. y = 5x - 7; (2, 3), (1,5), (-1, -12)

Let’s start with the ordered pair (2, 3).

Which number is the x value and which one is the y value? If you said x = 2 and y = 3, you are correct!

Let’s plug (2, 3) into the equation and see what we get:

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*Plug in 2 for x and 3 for y

This is a TRUE statement, so (2, 3) is a solution to the equation y = 5x - 7.

Now let’s take a look at (1, 5).

Which number is the x value and which one is the y value? If you said x = 1 and y = 5, you are right!

Let’s plug (1, 5) into the equation and see what we get:

*Plug in 1 for x and 5 for y

Whoops, it looks like we have ourselves a FALSE statement. This means that (1, 5) is NOT a solution to the

equation 5x - 7.

Now let’s look at (-1, -12).

Which number is the x value and which one is the y value? If you said x = -1 and y = -12, you are right!

Let’s plug (-1, -12) into the equation and see what we get:

*Plug in -1 for x and -12 for y

We have another TRUE statement. This means (-1, -12) is another solution to the equation y = 5x - 7.

Note that you were only given three ordered pairs to check, however, there are an infinite number of solutions

to this equation. It would very cumbersome to find them all.

Example 3: Determine whether each ordered pair is a solution of the given equation. ; (0, -3), (1,

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-3), (-1, -3)

Let’s start with the ordered pair (0, -3).

Which number is the x value and which one is the y value? If you said x = 0 and y = -3, you are correct!

Let’s plug (0, -3) into the equation and see what we get:

*Plug in 0 for x and -3 for y

This is a FALSE statement, so (0, -3) is NOT a solution to the equation

Now, let’s take a look at (1, -3).

Which number is the x value and which one is the y value? If you said x = 1 and y = -3, you are right!

Let’s plug (1, -3) into the equation and see what we get:

*Plug in 1 for x and -3 for y

This is a TRUE statement. This means that (1, -3) is a solution to the equation .

Now, let’s look at (-1, -3).

Which number is the x value and which one is the y value? If you said x = -1 and y = -3, you are right!

Let’s plug (-1, -3) into the equation and see what we get:

*Plug in -1 for x and -3 for y

This is a TRUE statement. This means that (1, -3) is a solution to the equation .

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Linear Equation in

Two Variables

Standard Form:

Ax + By = C

A linear equation in two variables is an equation that can be written in the form Ax + By = C, where A and B are not both0.

This form is called the standard form of a linear equation.

Example 4: Write the following linear equation in standard form.

y = 7x - 5

This means we want to write it in the form Ax + By = C.

*Inverse of add 7x is sub. 7x*In standard form

Graphing a Linear Equation

by Plotting Points

If the equation is linear:

Step 1: Find three ordered pair solutions.

You do this by plugging in ANY three values for x and find their corresponding y values.

Yes, it can be ANY three values you want, 1, -3, or even 10,000. Remember there are an

infinite number of solutions. As long as you find the corresponding y value that goes with

each x, you have a solution.

Step 2: Plot the points found in step 1.

Remember that each ordered pair corresponds to only one point on the graph.

The point lines up with both the x value of the ordered pair (x-axis) and the y value of the

ordered pair (y-axis).

Step 3: Draw the graph.

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A linear equation will graph as a straight line.

If you know it is a linear equation and your points don’t line up, then you either need to check

your math in step 1 and/or that you plotted all the points found correctly.

Example 5: Determine whether the equation is linear or not. Then graph the equation. y = 5x - 3

If we subtract 5x from both sides, then we can write the given equation as -5x + y = -3.

Since we can write it in the standard form, Ax + By = C, then we have a linear equation.

This means that we will have a line when we go to graph this.

Step 1: Find three ordered pair solutions.

I’m going to use a chart to organize my information. A chart keeps track of the x values that you are using and

the corresponding y value found when you used a particular x value.

If you do this step the same each time, then it will make it easier for you to remember how to do it.

I usually pick out three points when I know I’m dealing with a line. The three x values I’m going to use are -1,0, and 1. (Note that you can pick ANY three x values that you want. You do not have to use the values

that I picked.) You want to keep it as simple as possible. The following is the chart I ended up with after

plugging in the values I mentioned for x.

x y = 5x - 3 (x, y)

-1 y = 5(-1) - 3 = -8 (-1, -8)

0 y = 5(0) - 3 = -3 (0, -3)

1 y = 5(1) - 3 = 2 (1, 2)


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Graphing a Non-Linear Equation

by Plotting Points

If the equation is non-linear:

Step 1: Find six or seven ordered pair solutions.

Non-linear equations can vary on what the graph looks like. So it is good to have a lot of

points so you can get the right shape of the graph.



If the points line up draw a straight line through them. If the points are in a curve, draw a

curve through them.

Example 6: Determine whether the equation is linear or not. Then graph the equation.

If we subtract the x squared from both sides, we would end up with . Is this a linear equation?

Note how we have an x squared as opposed to x to the one power.

It looks like we cannot write it in the form Ax + By = C because the x has to be to the one power, not squared. So this is not a linear equation.

However, we can still graph it.

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The seven x values that I'm going to use are -3, -2, -1, 0, 1, 2, and 3. (Note that you can pick ANY x valuesthat you want. You do not have to use the values that I picked.) You want to keep it as simple as possible.

The following is the chart I ended up with after plugging in the values I mentioned for x.

Note that the carrot top ( )̂ represents an exponent. For example, x squared can be written as x^2. The second

column is showing you the 'scratch work' of how we got the corresponding value of y.

x (x, y)

-3 y = (-3)^2 - 4 = 5 (-3, 5)

-2 y = (-2)^2 - 4 = 0 (-2, 0)

-1 y = (-1)^2 - 4 = -3 (-1, -3)

0 y = (0)^2 - 4 = -4 (0, -4)

1 y = (1)^2 - 4 = -3 (1, -3)

2 y = (2)^2 - 4 = 0 (2, 0)

3 y = (3)^2 - 4 = 5 (3, 5)



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Example 7: Determine whether the equation is linear or not. Then graph the equation.

Do you think this equation is linear or not? It is a tricky problem because both the x and y variables are to the

one power. However, x is inside the absolute value sign and we can’t just take it out of there.

In other words, we can’t write it in the form Ax + By = C. This means that this equation is not a linearequation.

If you are unsure that an equation is linear or not, you can ALWAYS plug in x values and find the

corresponding y values to come up with ordered pairs to plot.


The seven x values that I'm going to use are -3, -2, -1, 0, 1, 2, and 3. (Note that you can pick ANY x valuesthat you want. You do not have to use the values that I picked.) You want to keep it as simple as possible.

The following is the chart I ended up with after plugging in the values I mentioned for x.

x (x, y)

-3 y = |-3 + 1| = 2 (-3, 2)

-2 y = |-2 + 1| = 1 (-2, 1)

-1 y = |-1 + 1| = 0 (-1, 0)

0 y = |0 + 1| = 1 (0, 1)

1 y = |1 + 1| = 2 (1, 2)

2 y = |2 + 1| = 3 (2, 3)

3 y = |3 + 1| = 4 (3, 4)



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Practice Problems

These are practice problems to help bring you to the next level. It will allow you to check and see if you have an

understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need

to practice it. Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to

get good at their sport or instrument. In fact there is no such thing as too much practice.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on

the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went

into finding that answer.

Practice Problem 1a: Plot the ordered pairs and name the quadrant or axis in which the point

lies.

1a. A(3, 1), B(-2, -1/2), C(2, -2), and D(0,1)(answer/discussion to 1a)

Practice Problem 2a: Determine if each ordered pair is a solution of the given equation.

2a. y = 4x - 10 (0, -10), (1, -14), (-1, -14)(answer/discussion to 2a)

Practice Problems 3a - 3c: Determine whether each equation is linear or not. Then graph the

equation.

3a. y = 2x - 13b.

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(answer/discussion to 3a) (answer/discussion to 3b)

3c. (answer/discussion to 3c)

Need Extra Help on these Topics?

The following are webpages that can assist you in the topics that were covered on this page:

http://www.purplemath.com/modules/plane.htm

This webpage helps you with plotting points.

http://www.math.com/school/subject2/lessons/S2U4L1DP.html

This website helps you with plotting points.

http://www.purplemath.com/modules/graphlin.htm

This webpage helps you with graphing linear equations.

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more

suggestions.


Last revised on July 3, 2011 by Kim Seward.

All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.

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Tutorial 13: Introduction to Functions


Learning Objectives


Know what a relation, function, domain and range is.1.

Find the domain and range of a relation.2.

Identify if a relation is a function or not.3.

Know how to use the vertical line test.4.

Evaluate functional values. 5.

Introduction

In this tutorial we will be taking a look at working with functions. We will start off by looking for the domain and range.

Then, we will move on to finding functional values. Don't be thrown by the different way function notation looks. When

you go to evaluate these functions, we are just plugging in values for x and simplifying, just like you have done a million

times before (it is one of those types of problems that I warn students to not make harder than it is). Just think of finding a

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functional value as a fancy assignment statement. I think we are ready to forge ahead.

Tutorial

Relation

A relation is a set of ordered pairs where the first components of the ordered pairs are the input values and the second

components are the output values.

Function

A function is a relation that assigns to each input number EXACTLY ONE output number.

Be careful. Not every relation is a function. A function has to fit the above definition to a tee.

Domain

The domain is the set of all input values to which the rule applies. These are called your independent variables. These are

the values that correspond to the first components of the ordered pairs it is associated with.

Range

The range is the set of all output values. These are called your dependent variables. These are the values that correspond

to the second components of the ordered pairs it is associated with.

Example 1: Find the domain and range of the relation. Also, determine whether the relation is a function. {(3,

2), (4, 3), (5, 4), (6, 5)}

DomainWe need to find the set of all input values. In terms of ordered pairs, that correlates with the first component of

each one. So, what do you get for the domain?

If you got {3, 4, 5, 6}, you are correct!

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RangeWe need to find the set of all output values. In terms of ordered pairs, that correlates with the second

component of each one. SO, what do you get for the range?

If you got {2, 3, 4, 5}, you are absolutely right!

Is this a function or not?We need to ask ourselves, does every first element (or input) correspond with EXACTLY ONE second

element (or output)? In this case, the answer is yes. 3 only goes with 2, 4 only goes with 3, 5 only goes with 4

and 6 only goes with 5.

So, this relation would be an example of a function.

Example 2: Find the domain and range of the relation. Also, determine whether the relation is a function.

Reading the ordered pairs off of the graph, we get

{(2, 3), (2, 4), (3, 3), (4, 3)}



If you got {2, 3, 4}, you are correct!

Note that if any value repeats, we only need to list it one time.


component of each one. So, what do you get for the range?

If you got {3, 4}, you are absolutely right!

Again, note that if any value repeats, we only need to list it one time.

Is this a function or not?We need to ask ourselves, does every first element (or input) correspond with exactly one second element (or

output)? In this case, the answer is NO. The input value of 2 goes with two output values, 3 and 4. It only

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takes one input value to associate with more than one output value to be invalid as a function.

In this case, the relation is not a function.

Example 3: Find the domain and range of the relation. Also, determine whether the relation is a function.

Reading the ordered pairs off of the diagram we get

{(a, 1), (b, 2), (c, 1), (d, 2)}



If you got {a, b, c, d}, you are correct!


component of each one. So, what do you get for the range?

If you got {1, 2}, you are absolutely right!

Is this a function or not?We need to ask ourselves, does every first element (or input) correspond with exactly one second element (or

output)? In this case, the answer is yes. a only goes with 1, b only goes with 1, c only goes with 2 and d only

goes with 2.

Note that a relation can still be a function if an output value associates with more than one input value as

shown in this example. But again, it would be a no no the other way around, where an input value

corresponds to two or more output values.

So, this relation would be an example of a function.

Example 4: Decide whether y is a function of x.

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To check if y is a function of x, we need to solve for y first and then check to see if there is only one output for

every input.

*Inverse of add x is sub. x

*Inverse of mult. by 5 is div. by 5

At this point we ask ourselves, would we get one value for y if you plug in any value for x?

If you answered yes, you are right on. For example, if we plugged in a 1 for x, then y would only equal one

value, 2/5.

Note that since it is solved for y, y is our output value and x is our input value.

Since our answer to that question is yes, that means by definition, y is a function of x.

Example 5: Decide whether y is a function of x.

To check if y is a function of x, we need to solve for y squared first and then check to see if there is only one

output for every input.


At this point we ask ourselves, would we get one value for y if you plug in any value for x?

If you answered no, you are correct.

For example, if our input value x is -16, then our output value y could either be 4 or - 4. Note that I could have

picked an infinite number of examples like this one. You only need to show one example where the input value

is associated with more than one output value to disqualify it from being a function.

This means that at least one input value is associated with more than one output value, so by definition, y

is not a function of x.

Vertical Line Test

If no vertical line can be drawn so that it intersects a graph more than once, then it is a graph of a function.

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Think about it, if a vertical line intersects a graph in more than one place, then the x value (input) would associate with more

than one y value (output), and you know what that means. The relation is not a function.

The next two examples illustrate this concept.

Example 6: Find the domain and range of each relation. Use the vertical line test to determine whether each

graph is a graph of a function.

DomainWe need to find the set of all input values. In terms of ordered pairs, that correlates with the first component

of each one. In terms of this two dimensional graph, that corresponds with the x values (horizontal axis).

Since that is the case, we need to look to the left and right and see if there are any end points. In this case,

note how the line has arrows at both ends, that means it would go on and on forever to the right and to the

left.

This means that the domain is {x | x is a real number}.


component of each one. In terms of this two dimensional graph, that corresponds with the y values (vertical

axis).

Since that is the case, we need to look up and down and see if there are any end points. In this case, note how

the line has arrows at both ends, that means it would go on and on forever up and down.

This means that the range is {y | y is a real number}.

Vertical Line TestThis graph would pass the vertical line test, because there would not be any place on it that we could draw a

vertical line and it would intersect it in more than one place.

Therefore, this is a graph of a function.

Example 7: Find the domain and range of each relation. Use the vertical line test to determine whether each

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graph is a graph of a function.

DomainWe need to find the set of all input values. In terms of ordered pairs, that correlates with the first component

of each one. In terms of this two dimensional graph, that corresponds with the x values (horizontal axis).

Since that is the case, we need to look to the left and right and see if there are any end points. In this case,

note that the farthest left point is (1, 0) and the farthest right point is (5, 0), and the circle is enclosed between

these values. That means that if we wrote out ordered pairs for all the values going around the circle, we would

only use the values from 1 to 5 for x.

This means that the domain is {x | 1 <x< 5}.


component of each one. In terms of this two dimensional graph, that corresponds with the y values (vertical

axis).

Since that is the case, we need to look up and down and see if there are any end points. In this case, note that

the highest point is (3, 2) and the lowest point is (3, -2), and the circle is enclosed between these values. That

means that if we wrote out ordered pairs for all the values going around the circle, we would only use the

values from -2 to 2 for y.

This means that the range is {y | -2 < y < 2}

Vertical Line TestThis graph would not pass the vertical line test because there is at least one place on it that we could draw a

vertical line and intersect it in more than one place. In fact, there are a lot of vertical lines that we can draw

that would intersect it in more than one place, but we only need to show one to say it is not a function.

The graph below shows one vertical line drawn through our graph that intersects it in two places: (3, 2) and (3,

-2). This shows that the input value of 3 associates with two output values, which is not acceptable in the

function world.

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Therefore, this is not a graph of a function.

Function Notation

f(x) read “f of x”

f is the function name. Output values are also called functional values. Note that you can use any letter to represent a

function name, f is a very common one used.

x is your input variable.

Think of functional notation as a fancy assignment statement. When you need to evaluate the function for a given value

of x, you simply replace x with that given value and simplify. Just like you do when you are finding the value of an

expression when you are given a number for your variable as shown in Tutorial 2: Algebraic Expressions. For

example, if we are looking for f(0), we would plug in 0 for the value of x in our function f.

Example 8: Find the functional values f(0), f(-1), and f(1) for the function .

Again, think of functional notation as a fancy assignment statement. For example, when we are looking for

f(0), we are going to plug in 0 for the value of x in our function f and so forth.

*Plug in 0 for x and evaluate

*Plug in -1 for x and evaluate


So our answers are f(0) = -1, f(-1) = -4, and f(1) = 2.

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Example 9: Find the functional values g(-1), g(1/2), and g(5) for the function .

*Plug in -1 for x and evaluate

*Plug in 1/2 for x and evaluate


So our answers are g(-1) = 4, g(1/2) = 1, and g(5) = 46.

Example 10: Find the functional values h(-1), h(1/2), and h(2) for the function h(x) = 3.

This is what is called a constant function. That means, no matter what the input value is, this functional

value is ALWAYS going to be 3.

Note how there is no input value visible. This is not the same as having 3x.

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*This is a constant function that is always 3



So our answers are h(-1) = 3, h(1/2) = 3, and h(2) = 3.

Practice Problems








Practice Problem 1a - 1b: Find the domain and range of the relation. Also, determine

whether the relation is a function.

1a. {(1, -1), (1, -2), (2, -3), (3, - 4)}(answer/discussion to 1a)

1b.

(answer/discussion to 1b)

Practice Problems 2a - 2b: Find the domain and range of each relation. Use the vertical line

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test to determine whether each graph is a graph of a function.

2a.

(answer/discussion to 2a)

2b.

(answer/discussion to 2b)

Practice Problems 3a - 3b: Decide whether y is a function of x.

3a. (answer/discussion to 3a)

3b. 2x - 3y = 7(answer/discussion to 3b)

Practice Problems 4a - 4b: Find the functional values.

4a. f(x) = 5x + 2 f(-1), f(0), f(.5)(answer/discussion to 4a)

4b. g(x) = 2 g(-1), g(0), g(.4)(answer/discussion to 4b)



http://www.purplemath.com/modules/fcns.htm

This website goes over what a function is and what domain and range are.

http://www.purplemath.com/modules/fcnnot.htm

This website goes over function notation.

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suggestions.








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Tutorial 14: Graphing Linear Equations


Learning Objectives


Find the x- and y-intercepts of a linear function.1.

Graph a linear function using the x- and y-intercepts.2.

Graph vertical and horizontal lines. 3.

Introduction

In Tutorial 12: Graphing Equations, we went over graphing in general, learning the basics of how to graph ANY equation

by plotting points. In this tutorial we will be looking specifically at graphing lines. The concept of using intercepts to help

graph will be introduced on this page, as well as vertical and horizontal lines. Actually, the process of graphing by plotting

points and graphing by using intercepts are essentially the same. Intercepts are just special types of solutions, but solutions

none the less. So once we find them, we plot them just the same as any other ordered pair that is a solution. Once we plot

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them, we draw our graph in the same fashion as when we had non-intercept points. So, basically, when you graph, you plot

solutions (whether they are intercept points or not) and connect the dots to get your graph.

Tutorial

x-intercept

The x-intercept is where the graph crosses the x axis.

The word 'intercept' looks like the word 'intersect'. Think of it as where the graph intersects the x-axis.

With that in mind, what value is y always going to be on the x-intercept? No matter where you are on the x-axis, y’s

value is 0, that is a constant. We will use that bit of information to help us find the x-intercept when given an equation.

y-intercept

If the x-intercept is where the graph crosses the x-axis where do you think the graph crosses for the y-intercept? If you

said the y-axis, you are absolutely right.

This time it is x’s value that is 0. Any where you would cross the y-axis, x’s value is always 0. We will use this tidbit to

help us find the y-intercept when given an equation.

Below is an illustration of a graph of a linear function which highlights the x and y intercepts:

In the above illustration, the x-intercept is the point (2, 0) and the y-intercept is the point (0, 3).

Keep in mind that the x- and y- intercepts are two separate points. There is only one point that can be both an x- and

y- intercept at the same time, do you know what point that is?

If you said the origin (0, 0), give yourself a pat on the back.

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Sketching a Graph

Using Intercepts

Step 1: Find the x- and y- intercepts.

You find the x-intercept by plugging in 0 for y and solving for x.

You find the y-intercept by plugging in 0 for x and solving for y.

Step 2: Find at least one more point.

You do this by plugging in ANY value(s) for x and finding it's corresponding y value.

This is just like we showed you in Tutorial 12: Graphing Equations.

Step 3: Plot the intercepts and point(s) found in steps 1 and 2.

Remember that intercepts are points on the graph, too. They are plotted just like any other point.


The graph of a linear function is a straight line.

Example 1: Graph each linear function by finding the x- and y- intercepts. y= 5 - 3x


Let’s first find the x-intercept.

What value are we going to use for y?

You are correct if you said y = 0.

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*Find x-int. by replacing y with 0*Inverse of add 5 is sub. 5

*Inverse of mult. by -3 is div. by -3

The x-intercept is (5/3, 0).

Next, we will find the y- intercept.

What value are we going to plug in for x?

If you said x = 0 you are right.

*Find y-int. by replacing x with 0

The y-intercept is (0, 5)


We can plug in any x value we want as long as we get the right corresponding y value and the function existsthere.

Let’s put in an easy number x = 1:

*Replace x with 1

So the ordered pair (1, 2) is another solution to our function.

Note that we could have plugged in any value for x: 5, 10, -25, ..., but it is best to keep it as simple as possible.

The solutions that we found are:

x y (x, y)

5/3 0 (5/3, 0)

0 5 (0, 5)

1 2 (1, 2)


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Example 2: Graph each linear function by finding the x- and y- intercepts. -3x = y


Let’s first find the x-intercept.

What value are we going to use for y?

You are correct if you said y = 0.

*Find x-int. by replacing y with 0*Inverse of mult. by -3 is div. by -3

The x-intercept is (0, 0).

Next, we will find the y- intercept.

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What value are we going to plug in for x?

If you said x = 0, you are right.

*Find y-int. by replacing x with 0

The y-intercept is (0, 0)

Hey, look at that, we ended up with the exact same point for both our x- and y-intercepts. As mentioned

above, there is only one point that can be both an x- and y- intercept at the same time, the origin (0, 0).


Since we really have found only one point this time, we better find two additional solutions so we have a

total of three points.

We can plug in any x value we want as long as we get the right corresponding y value and the function existsthere.

Let’s put in an easy number x = 1:

*Replace x with 1

So the ordered pair (1, -3) is another solution to our function.

Let’s put in another easy number x = -1:

*Replace x with -1

So the ordered pair (-1, 3) is another solution to our function.

Note that we could have plugged in any value for x: 5, 10, -25, ..., but it is best to keep it as simple as possible.


x y (x, y)

0 0 (0, 0)

1 -3 (1, -3)

-1 3 (-1, 3)

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Vertical Lines

x = c

If you have an equation x = c, where c is a constant, and you are wanting to graph it on a two dimensional graph, this would

be a vertical line with x-intercept of (c, 0).

Even though you do not see a y in the equation, you can still graph it on a two dimensional graph. Remember that the graph

is the set of all solutions for a given equation. If all the points are solutions then any ordered pair that has an x value of c

would be a solution. As long as x never changes value, it is always c, then you have a solution. In that case, you will endup with a vertical line.

Below is an illustration of a vertical line x = c:

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Horizontal Lines

y = c

If you have an equation y = c, where c is a constant, and you are wanting to graph it on a two dimensional graph, this would

be a horizontal line with y- intercept of (0, c).

Even though you do not see an x in the equation, you can still graph it on a two dimensional graph. Remember that the graph

is the set of all solutions for a given equation. If all the points are solutions then any ordered pair that has an y value of cwould be a solution. As long as y never changes value, it is always c, then you have a solution. In that case, you will end up

with a horizontal line.

Below is an illustration of a horizontal line y = c:

Example 3: Graph the linear equation y = 4.

It looks like it fits the form y = c.With that in mind, what kind of line are we going to end up with?

Horizontal.

Note how the directions did not specify that we had to use intercepts to do our graph. Any time you take

a math test or do homework, make sure that you follow directions carefully. If it specifies a certain way

to do a problem, then you need to follow that plan (like in the above examples 1 and 2). If it does notspecify, like in this example, then you can use what ever “legitimate” way works to get the job done.

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AND


Since this is a special type of line, I thought I would talk about steps 1 and 2 together.

It doesn’t matter what x is, y is always 4. So for our solutions we just need three ordered pairs such that y = 4.

Note that the y-intercept (where x = 0) is at (0, 4).

Do we have a x-intercept? The answer is no. Since y has to be 4, then it can never equal 0, which is the

criteria of an x-intercept. Also, think about it, if we have a horizontal line that crosses the y-axis at 4, it will

never ever cross the x-axis.

So, some points that we can use are (0, 4), (1, 4) and (2, 4). These are all ordered pairs that fit the

criteria of y having to be 4.

Of course, we could have used other solutions, there are an infinite number of them.


x y (x, y)

0 4 (0, 4)

1 4 (1, 4)

2 4 (2, 4)



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Example 4: Graph the linear equation x + 3 = 0.

Note how if we subtract 3 from both sides, we can write this as x = -3, which means it can be written in the

form x = c. So, what type of line are we going to end up with?

Vertical.


AND


Since this is a special type of line, I thought I would talk about steps 1 and 2 together.

It does not matter what y is, as long as x is -3.

Note that the x-intercept is at (-3, 0).

Do we have a y-intercept? The answer is no. Since x can never equal 0, then there will be no y-intercept forthis equation.

Some points that would be solutions are (-3, 0), (-3, 1), and (-3, 2).

Again, I could have picked an infinite number of solutions.


x y (x, y)

-3 0 (-3, 0)

-3 1 (-3, 1)

-3 2 (-3, 2 )


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Practice Problems








Practice Problems 1a - 1b: Graph each linear function by finding x- and y-intercepts.

1a. 2x - 3y = -6(answer/discussion to 1a)

1b. x = 3y(answer/discussion to 1b)

Practice Problems 2a - 2b: Graph each linear equation.

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2a. x = 4(answer/discussion to 2a)

2b. y + 5 = 0(answer/discussion to 2b)


The following is a webpage that can assist you in the topics that were covered on this page:

http://www.purplemath.com/modules/intrcept.htm

This website goes over x- and y-intercepts.


suggestions.








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b b 2 c b g Title

|



Tutorial 15: The Slope of a Line


Learning Objectives


Find the slope given a graph, two points or an equation.1.

Write a linear equation in slope/intercept form.2.

Determine if two lines are parallel, perpendicular, or neither. 3.

Introduction

This tutorial takes us a little deeper into linear equations. We will be looking at the slope of a line. We will also look at the

relationship between the slopes of parallel lines as well as perpendicular lines. Let's see what you can do with slopes.

1 of 12

Tutorial

Slope

The slope of a line measures the steepness of the line.

Most of you are probably familiar with associating slope with "rise over run".

Rise means how many units you move up or down from point to point. On the graph that would be a change in

the y values.

Run means how far left or right you move from point to point. On the graph, that would mean a change of xvalues.

Here are some visuals to help you with this definition:

Positive slope:

Note that when a line has a positive slope it goes up left to right.

Negative slope:

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Note that when a line has a negative slope it goes down left to right.

Zero slope:

slope = 0

Note that when a line is horizontal the slope is 0.

Undefined slope:

slope = undefined

Note that when the line is vertical the slope is undefined.

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Slope Formula Given Two Points

Given two points and

The subscripts just indicate that these are two different points. It doesn't matter which one you call point 1 and which one

you call point 2 as long as you are consistent throughout that problem.

Note that we use the letter m to represent slope.

Example 1: Find the slope of the straight line that passes through (-5, 2) and (4, -7).

*Plug in x and y values into slope formula

*Simplify

Make sure that you are careful when one of your values is negative and you have to subtract it as we didin line 2. 4 - (-5) is not the same as 4 - 5.

The slope of the line is -1.

Example 2: Find the slope of the straight line that passes through (1, 1) and (5, 1).


*Simplify

It is ok to have a 0 in the numerator. Remember that 0 divided by any non-zero number is 0.

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The slope of the line is 0.



*Simplify

Since we did not have a change in the x values, the denominator of our slope became 0. This means that wehave an undefined slope. If you were to graph the line, it would be a vertical line, as shown above.

The slope of the line is undefined.

Slope/Intercept Equation of a Line

If your linear equation is written in this form, m represents the slope and b represents the y-intercept.

This form can be handy if you need to find the slope of a line given the equation.

Function Notation of the


m still represents slope and b still represents the y-intercept.

Example 4: Find the slope and the y-intercept of the line .

As mentioned above, if the equation is in the slope/intercept form, we can easily see what the slope and

y-intercept are.

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Let’s go ahead and get it into the slope/intercept form first:

*Sub. 3x and add 6 to both sides


*Written in slope/intercept form

Lining up the form with the equation we got, can you see what the slope and y-intercept are?

In this form, the slope is m, which is the number in front of x. In our problem, that would have to be -1.

In this form, the y-intercept is b, which is the constant. In our problem, that would be 2.

The answer is the slope is -1 and the y-intercept is 2.


This example is written in function notation, but is still linear. As shown above, you can still read off the slope

and intercept from this way of writing it.

In this example, it is already written in the slope/intercept form, so we do not have to mess around with it. We

can get down to business and answer our question of what are the slope and y-intercept.



In this form, the slope is m, which is the number in front of x. In our problem, that would have to be 2.

In this form, the y-intercept is b, which is the constant. In our problem, that would be -1.

The answer is the slope is 2 and the y-intercept is -1.

Example 6: Find the slope and the y-intercept of the line x = 5.

Note how we do not have a y. This type of linear equation was shown in Tutorial 14: Graphing Linear

Equations. When we have x = c, where c is a constant, then this graph is what type of line?

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If you said vertical, you are correct.

Since this is a special type of linear equation that can’t be written in the slope/intercept form, I’m going to give

you a visual of what is happening and then from that let’s see if we can’t figure out the slope and y-intercept.

The graph would look like this:

First, let’s talk about the slope. Note that all the x values on this graph are 5. That means the change in x,which is the denominator of the slope formula, would be 5 - 5 = 0. Well you know that having a 0 in the

denominator is a big no, no. This means the slope is undefined. As shown above, whenever you have a

vertical line your slope is undefined.

Now let’s look at the y-intercept. Looking at the graph, you can see that this graph never crosses the y-axis,

therefore there is no y-intercept either. Another way to look at this is the x value has to be 0 when looking for

the y-intercept and in this problem x is always 5.

So, for all our efforts on this problem, we find that the slope is undefined and the y-intercept does not

exist.

Example 7: Find the slope and the y-intercept of the line y = -2.

Note how we do not have an x. This type of linear equation was shown in Tutorial 14: Graphing Linear

Equations. When we have y = c, where c is a constant, then this graph is what type of line?If you said horizontal, you are correct.




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First, let’s talk about the slope. Note how all of the y values on this graph are -2. That means the change in y,which is the numerator of the slope formula would be -2 - (-2) = 0. Having 0 in the numerator and a non-zero

number in the denominator means only one thing. The slope equals 0.

Now let’s look at the y-intercept. Looking at the graph, you can see that this graph crosses the y-axis at (0,-2). So the y-intercept is (0, -2).

The slope is 0 and the y-intercept is -2.

Parallel Lines and Their Slopes

In other words, the slopes of parallel lines are equal.

Note that two lines are parallel if there slopes are equal and they have different y-intercepts.

Perpendicular Lines and Their Slopes

In other words, perpendicular slopes are negative reciprocals of each other.

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Example 8: Determine if the lines are parallel, perpendicular, or neither. and .

In order for these lines to be parallel their slopes would have to be equal and to be perpendicular they would

have to be negative reciprocals of each other.

So let’s find out what the slopes are. Since the equations are already in the slope/intercept form, we can look at

them and see the relationship between the slopes. What do you think? The slope of the first equation is 7

and the slope of the second equation is 7.

Since the two slopes are equal and their y-intercepts are different, the two lines would have to be

parallel.

Example 9: Determine if the lines are parallel, perpendicular, or neither. and

.

Again, the equations are already in the slope/intercept form, so let’s go right to looking for the slope. What did

you find?

I found that the slope of the first equation is 4 and the slope of the second equation is -1/4. So what does

that mean?

Since the two slopes are negative reciprocals of each other, the two lines would be perpendicular to eachother.


Writing the first equation in the slope/intercept form we get:

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*Inverse of add 10x is sub. 10x*Written in slope/intercept form

Writing the second equation in the slope/intercept form we get:

*Inverse of add 4x is sub. 4x




have to be negative reciprocals of each other. So let’s find out what the slopes are. Since the equations are

now in the slope/intercept form, we can look at them and see the relationship between the slopes. What do you

think?

The slope of the first equation is -10 and the slope of the second equation is -2.

Since the two slopes are not equal and are not negative reciprocals of each other, then the answer would

be neither.

Practice Problems



to practice it. Even the best athletes and musicians had help along the way and lots of practice, practice, practice, toget good at their sport or instrument. In fact there is no such thing as too much practice.




Practice Problem 1a - 1b: Find the slope of the straight line that passes through the given

points.

1a. (3, 5) and (-1, -8)(answer/discussion to 1a)

1b. (4, 2) and (4, -2)(answer/discussion to 1b)

Practice Problems 2a - 2c: Find the slope and the y-intercept of the line.

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2b. x = -2(answer/discussion to 2b)

2c. y = -1(answer/discussion to 2c)

Practice Problems 3a - 3b: Determine if the lines are parallel, perpendicular, or neither.

3a. and (answer/discussion to 3a)

3b. and (answer/discussion to 3b)

Practice Problem 4a: Determine the slope of the line.

4a.




http://www.purplemath.com/modules/slope.htm

This webpage helps you with slope.


11 of 12

This website covers slopes and y-intercept.

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some moresuggestions.








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b b 2 c b g Title

|



Tutorial 15: The Slope of a Line


Learning Objectives


Find the slope given a graph, two points or an equation.1.

Write a linear equation in slope/intercept form.2.

Determine if two lines are parallel, perpendicular, or neither. 3.

Introduction

This tutorial takes us a little deeper into linear equations. We will be looking at the slope of a line. We will also look at the

relationship between the slopes of parallel lines as well as perpendicular lines. Let's see what you can do with slopes.

1 of 12

Tutorial

Slope

The slope of a line measures the steepness of the line.

Most of you are probably familiar with associating slope with "rise over run".

Rise means how many units you move up or down from point to point. On the graph that would be a change in

the y values.

Run means how far left or right you move from point to point. On the graph, that would mean a change of xvalues.

Here are some visuals to help you with this definition:

Positive slope:

Note that when a line has a positive slope it goes up left to right.

Negative slope:

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Note that when a line has a negative slope it goes down left to right.

Zero slope:

slope = 0

Note that when a line is horizontal the slope is 0.

Undefined slope:

slope = undefined

Note that when the line is vertical the slope is undefined.

3 of 12

Slope Formula Given Two Points

Given two points and

The subscripts just indicate that these are two different points. It doesn't matter which one you call point 1 and which one

you call point 2 as long as you are consistent throughout that problem.

Note that we use the letter m to represent slope.

Example 1: Find the slope of the straight line that passes through (-5, 2) and (4, -7).


*Simplify

Make sure that you are careful when one of your values is negative and you have to subtract it as we didin line 2. 4 - (-5) is not the same as 4 - 5.

The slope of the line is -1.



*Simplify

It is ok to have a 0 in the numerator. Remember that 0 divided by any non-zero number is 0.

4 of 12

The slope of the line is 0.



*Simplify

Since we did not have a change in the x values, the denominator of our slope became 0. This means that wehave an undefined slope. If you were to graph the line, it would be a vertical line, as shown above.

The slope of the line is undefined.


If your linear equation is written in this form, m represents the slope and b represents the y-intercept.

This form can be handy if you need to find the slope of a line given the equation.

Function Notation of the


m still represents slope and b still represents the y-intercept.


As mentioned above, if the equation is in the slope/intercept form, we can easily see what the slope and

y-intercept are.

5 of 12

Let’s go ahead and get it into the slope/intercept form first:

*Sub. 3x and add 6 to both sides




In this form, the slope is m, which is the number in front of x. In our problem, that would have to be -1.

In this form, the y-intercept is b, which is the constant. In our problem, that would be 2.

The answer is the slope is -1 and the y-intercept is 2.


This example is written in function notation, but is still linear. As shown above, you can still read off the slope

and intercept from this way of writing it.

In this example, it is already written in the slope/intercept form, so we do not have to mess around with it. We

can get down to business and answer our question of what are the slope and y-intercept.



In this form, the slope is m, which is the number in front of x. In our problem, that would have to be 2.

In this form, the y-intercept is b, which is the constant. In our problem, that would be -1.

The answer is the slope is 2 and the y-intercept is -1.

Example 6: Find the slope and the y-intercept of the line x = 5.

Note how we do not have a y. This type of linear equation was shown in Tutorial 14: Graphing Linear

Equations. When we have x = c, where c is a constant, then this graph is what type of line?

6 of 12

If you said vertical, you are correct.




First, let’s talk about the slope. Note that all the x values on this graph are 5. That means the change in x,which is the denominator of the slope formula, would be 5 - 5 = 0. Well you know that having a 0 in the

denominator is a big no, no. This means the slope is undefined. As shown above, whenever you have a

vertical line your slope is undefined.

Now let’s look at the y-intercept. Looking at the graph, you can see that this graph never crosses the y-axis,

therefore there is no y-intercept either. Another way to look at this is the x value has to be 0 when looking for

the y-intercept and in this problem x is always 5.

So, for all our efforts on this problem, we find that the slope is undefined and the y-intercept does not

exist.

Example 7: Find the slope and the y-intercept of the line y = -2.

Note how we do not have an x. This type of linear equation was shown in Tutorial 14: Graphing Linear

Equations. When we have y = c, where c is a constant, then this graph is what type of line?If you said horizontal, you are correct.




7 of 12

First, let’s talk about the slope. Note how all of the y values on this graph are -2. That means the change in y,which is the numerator of the slope formula would be -2 - (-2) = 0. Having 0 in the numerator and a non-zero

number in the denominator means only one thing. The slope equals 0.

Now let’s look at the y-intercept. Looking at the graph, you can see that this graph crosses the y-axis at (0,-2). So the y-intercept is (0, -2).

The slope is 0 and the y-intercept is -2.

Parallel Lines and Their Slopes

In other words, the slopes of parallel lines are equal.

Note that two lines are parallel if there slopes are equal and they have different y-intercepts.

Perpendicular Lines and Their Slopes

In other words, perpendicular slopes are negative reciprocals of each other.

8 of 12



have to be negative reciprocals of each other.

So let’s find out what the slopes are. Since the equations are already in the slope/intercept form, we can look at

them and see the relationship between the slopes. What do you think? The slope of the first equation is 7

and the slope of the second equation is 7.

Since the two slopes are equal and their y-intercepts are different, the two lines would have to be

parallel.

Example 9: Determine if the lines are parallel, perpendicular, or neither. and

.

Again, the equations are already in the slope/intercept form, so let’s go right to looking for the slope. What did

you find?

I found that the slope of the first equation is 4 and the slope of the second equation is -1/4. So what does

that mean?

Since the two slopes are negative reciprocals of each other, the two lines would be perpendicular to eachother.


Writing the first equation in the slope/intercept form we get:

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*Inverse of add 10x is sub. 10x*Written in slope/intercept form

Writing the second equation in the slope/intercept form we get:





have to be negative reciprocals of each other. So let’s find out what the slopes are. Since the equations are

now in the slope/intercept form, we can look at them and see the relationship between the slopes. What do you

think?

The slope of the first equation is -10 and the slope of the second equation is -2.

Since the two slopes are not equal and are not negative reciprocals of each other, then the answer would

be neither.

Practice Problems



to practice it. Even the best athletes and musicians had help along the way and lots of practice, practice, practice, toget good at their sport or instrument. In fact there is no such thing as too much practice.




Practice Problem 1a - 1b: Find the slope of the straight line that passes through the given

points.

1a. (3, 5) and (-1, -8)(answer/discussion to 1a)

1b. (4, 2) and (4, -2)(answer/discussion to 1b)

Practice Problems 2a - 2c: Find the slope and the y-intercept of the line.

10 of 12


2b. x = -2(answer/discussion to 2b)

2c. y = -1(answer/discussion to 2c)

Practice Problems 3a - 3b: Determine if the lines are parallel, perpendicular, or neither.

3a. and (answer/discussion to 3a)

3b. and (answer/discussion to 3b)

Practice Problem 4a: Determine the slope of the line.

4a.




http://www.purplemath.com/modules/slope.htm

This webpage helps you with slope.


11 of 12

This website covers slopes and y-intercept.









12 of 12

b b 2 c b g Title

|



Tutorial 16: Equations of Lines


Learning Objectives


Use the slope/intercept form to write a linear equation given the slope and y-intercept.1.

Use the slope/intercept form to graph a linear equation.2.

Use the point/slope equation to set up an equation given any point on the line and the slope.3.

Use the point/slope equation to set up an equation given two points on the line.4.

Use the point/slope equation to set up an equation given a point on the line and a parallel line.5.

Use the point/slope equation to set up an equation given a point on the line and a perpendicular line. 6.

Introduction

In this tutorial we will dive even deeper into linear equations. We will be going over how to come up with our own equations

given certain information. After you finish tutorials 12 - 17, you will be an old pro at linear equations and graphing. Let's

see what we can do with slopes, y-intercepts and linear equations.

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Tutorial

We are going to revisit the slope/intercept form of the line for a moment. This was initially introduced in Tutorial 15: Slope

of a Line. We are going to use it again to help us come up with equations of lines as well as give us another way to graph

lines.


OR

In this form, m represents the slope and b represents the y-intercept of the line.

Example 1: Use the slope/intercept form of the linear equation to write the equation of a line with slope - 4; y-intercept (0, ½).

Since we have the two missing pieces of the puzzle, the slope and the y-intercept, then we can go right into

plugging in - 4 for m (slope) and ½ for b (y-intercept) into the slope/intercept form and we will have our

equation.

*Slope/intercept form of a line

y = -4x + 1/2 is the equation of the line that has a slope of - 4 and y-intercept of (0, ½).

Nothing to it.

Graphing a Linear Equation

Using the Slope/Intercept Form

Step 1: Write the linear equation in the slope/intercept form if necessary.

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Some equations will already be in the slope/intercept form. In that case, you do not have to rewrite it.

However, if it isn't in that form, then you need to rewrite it. Basically, to get it into the slope/intercept form,

you solve the linear equation for y.

Step 2: Identify the slope and y-intercept of the linear equation.

This is done exactly as shown in Tutorial 15: The Slope of the Line, where m is the slope and b is the

y-intercept when the linear equation is written in the form y = mx + b.

Step 3: Plot the y-intercept point on the graph.

Remember, that when you identify b as your y-intercept that the ordered pair for it is (0, b).

Recall in Tutorial 14: Graphing Linear Equations that the x value on the y-intercept is always 0.

Step 4: Use the slope to find a second point on the graph.

Recall in Tutorial 15: The Slope of the Line that the phrase "rise over run" is used in conjunction with

slope.

Positive SlopeIf your slope is positive, then you are going to go in the same direction for both the rise and the run. In

other words, you either go in a positive direction for rise (up) and a positive direction for run (right) OR a

negative direction for rise (down) and a negative direction for run (left).

For example, if the slope is 2/3, then you can rise up 2 and run right 3 from any point that is on the line. Or,

you can go down 2 and run left 3 from any point that is on the line.

Negative SlopeIf your slope is negative, then you are going to go in opposite directions for the rise and the run. In other

words, you either go in a negative direction for rise (down) and a positive direction for run (right) OR a positive

direction for rise (up) and a negative direction for run (left).

For example, if the slope is -2/3, then you can go down 2 and run right 3 from any point that is on the line. Or,

you can rise 2 and run left 3 from any point that is on the line.

Integer SlopesIf your slope is an integer, remember that the denominator is understood to be 1. So, the run part of an integer

slope is going to be 1. For example, if the slope is 3, you want to think of it as 3/1. You would rise up 3 and

run right 1 from any point on the line to get another point on the line.


Note that this gives us another way to graph linear equations. Tutorial 12: Graphing Equations covers graphing by

plotting points, Tutorial 14: Graphing Linear Equation covers graphing using intercepts. This tutorial covers graphing

using the slope and the y-intercept. It doesn’t matter which method that you use, they all get the job done. However, if a

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test, homework or an instructor asks you to use a certain method make sure you follow the directions and show the

work for the asked method.

Let’s look at some examples for this new method.

Example 2: Graph the linear equation using the point/slope form.

3x - 2y = 4

Step 1: Write the linear equation in the slope/intercept form, if necessary.



*Slope/intercept form of the line


Looking at this equation and lining it up with the slope/intercept form, what do you get for the slope and

y-intercept?

I got m = 3/2 and y-intercept = -2.



The slope is 3/2.

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Starting on the y-intercept (0, -2), we will rise up 3 and run right 2.


Example 3: Graph the linear equation using the point/slope form.

y = -x

Step 1: Write the linear equation in the slope/intercept form, if necessary.

This linear equation is already in the slope/intercept form.



Lining up everything, what do you get for the slope and the y-intercept??

Note how we are missing a constant being added to the x term. If we are missing that constant, what is it

understood to be???

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The slope is -1 and the y-intercept is 0.



The slope is -1 or -1/1.

Starting on the y-intercept (0, 0), we will go down 1 and run right 1.


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Point/Slope Form of an Equation

A line going through

the point and

having slope of m

would have the equation

In example 1 above, the information was given to us in a nice neat little package, all ready for us to directly plug it into the

slope/intercept form. As you know, life isn’t always that nice. Sometimes we need to work with the information first before

we can write up our linear equation. Sometimes we need the help of the point/slope equation when we are not given the

y-intercept specifically.

We can use this form when we need to come up with a linear equation and we don’t know the y-intercept.

No matter what form that you end up using, keep in mind that you ALWAYS need two pieces of information when you go

to write an equation:

ANY point on the line1.

Slope2.

Example 4: Find the equation of the line with the given slope and containing the given point. Write the

equation in slope/intercept form.

Passes through (1, 7) and has a slope of -2.

What are the two things we need to write an equation of a line????

If you said any point on the line and the slope, you are correct.

Looks like we have all the information we need. We are ready to put our equation together. Since we don’t

have the y-intercept, we will have to use the point/slope form since that is set up for ANY point (not just the

y-int.).

*Point/slope form of the line

Next, we want to write it in the slope/intercept form, which basically means we need to solve for y.

*Dist. the -2 through ( )

*Inverse of sub. 7 is add 7


The equation of the line that passes through (1, 7)and has a slope of -2 is y = -2x + 9.

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Example 5: Find the equation of a line passing through the given points. Use function notation to write the

equation.

(-5, 3) and (-1, 4).

What are the two things we need to write an equation of a line????If you said any point on the line and the slope, you are correct.

We have more than enough points. However, what about the slope? Does this mean we can’t work out the

problem? You are not going to get off that easily. We do have a way of finding the slope. Tutorial 15: The

Slope of a Line shows us how we can get the slope given two points.

Let’s find that slope:

*Slope formula

*Plug in values

*Simplify

OK, now we have our slope, which is 1/4. Now it is just like example 4 above, we want to put the slope and

one point into the point/slope equation.


The directions said to write it using function notation. This means we want to solve it for y (get it in the

slope/int form) and then put it in function notation.

*Dist. 1/4 through ( )


*LCD is 4

*Mult. 3/1 by 4/4 to get 12/4


*Function notation

The line that passes through (-5, 3) and (-1, 4) in function notation is f(x) = 1/4 x + 17/4.

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Example 6: Find the equation of the line. Write the equation using function notation. Passes through (2, 3)

and parallel to .

What are the two things we need to write an equation of a line????If you said any point on the line and the slope you are correct.

We have our point. However, what about the slope?

We need to do a little digging to get it.

Recall that Tutorial 15: The Slope of a Line tells us that parallel lines have the same slope. So, if we know

the slope of the line parallel to our line, we have it made.

Find the slope of the parallel line:


Now, keep in mind that this is not the equation of our line but of the line parallel to our line. We needed to

write it this way so we could get the slope. And it looks like the slope is -3. And since our line is parallel to a

line that has a slope of -3, our line also has a slope of -3.

OK, now we have our slope, which is -3. Now it is just like examples 4 and 5 above, we want to put the slope

and one point into the point/slope equation.




*Dist. -3 through ( )*Inverse of sub. 3 is add 3


*Function notation

The line that passes through (2, 3) and is parallel to y = 5 - 3x is f(x) = -3x + 9.

Example 7: Find the equation of the line. Write the equation using function notation. Passes through (-1, 3)

and perpendicular to .

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What are the two things we need to write an equation of a line????

If you said any point on the line and the slope, you are correct.

We have our point. However, what about the slope?

We need to do a little work in that department.

Recall that Tutorial 15: The Slope of a Line tells us that the slopes of perpendicular lines are negative

reciprocals of each other. So, if we know the slope of the line perpendicular to our line, we have it made.

Find the slope of the perpendicular line:




In this form, we can tell that the slope of this line is ½. Since our line is perpendicular to this, we need to take

the negative reciprocal of ½ to get our slope.

What did you come up with?

I came up with -2 for the slope of our line.

Now we can go on to the equation of our line:




*Dist. -2 through the ( )


*Slope/intercept form of the line*Function notation

The line that passes through (-1, 3) and is perpendicular to x - 2y = 5 is f(x) = -2x + 1.

Example 8: Write an equation of the line that is horizontal and passes through (4, -1).

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Recall from Tutorial 14: Graphing Equations that a horizontal line is of the form y = c. It is one of our

‘special’ types of lines that does not have both x and y showing in the equation. So, we will not use the

point/slope form, but go right into setting up the equation y = c.

Since it passes through (4, -1), and a horizontal line is in the form y = c, where the y value is ALWAYS equal to

the same value throughout, this means our equation would have to be y = -1.

Note that -1 is the y value of the ordered pair given.

Example 9: Write an equation of the line that has an undefined slope and passes through (2, 3).

Recall from Tutorial 14: Graphing Equations and Tutorial 15: The Slope of the Line that the ‘special’ type

of line that has an undefined slope is a vertical line and a vertical line is of the form x = c. So, we will not use

the point/slope form, but go right into setting up the equation x = c.

Since it passes through (2, 3), and a vertical line is in the form x = c, where the x value is ALWAYS equal to the

same value throughout, this means our equation would have to be x = 2.

Note that 2 is the x value of the ordered pair given.

Practice Problems








Practice Problem 1a: Use the slope-intercept form of the linear equation to write the equation

of the line.

1a. Slope 2/3; y-intercept (0, -2)(answer/discussion to 1a)

Practice Problem 2a: Graph the linear equation using the point/slope form.

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2a. 3x + 5y = 10(answer/discussion to 2a)

Practice Problem 3a: Find the equation of the line with the given slope and containing the

given point. Write the equation in slope-intercept form.

3a. Passes through (-3, 2) and has a slope of -1/2.(answer/discussion to 3a)

Practice Problem 4: Find the equation of a line passing through the given points. Use function

notation to write the equation.

4a. (0, 0) and (5, 10)(answer/discussion to 4a)

Practice Problems 5a - 5b: Write an equation of the line.

5a. Slope 0; passes through (1, 2)(answer/discussion to 5a)

5b. Vertical; passes through (-1, -2)(answer/discussion to 5b)

Practice Problems 6a - 6b: Find an equation of the line. Write the equation using function

notation.

6a.Passes through (2, 3) and parallel to 5x + 2y = 4(answer/discussion to 6a)

6b. Passes through (-1, 0) and perpendicular to 3x - y = 2(answer/discussion to 6b)



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This website helps you with graphing linear equations.

http://www.purplemath.com/modules/slopgrph.htm

This website covers graphing linear equations using slopes.

http://www.purplemath.com/modules/slopyint.htm

This website goes over the meaning of slope and y-intercept.

http://www.purplemath.com/modules/strtlneq.htm

This website helps you with writing linear equations.









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