Graphing Linear Functions

1. graph linear functions.

2. write equations in standard form.

Warm Up, Do Now!

• Solve for y

1.3 – y = 10

2.2y + 4 = 8

3.9y + (-1) = 8

4.0 – 5 = -10y

Practice1) Solve for y: 2x + y = 4

2) Solve for y: 4x + 2y = -63) Solve for y: x – 3y = 6

Solve for y means isolate y. Get y all by itself!

1) Review: Solve for y 2x + y = 4

• Draw “the river”

• Subtract 2x from both sides

- 2x - 2x

y = -2x + 4

2) Solve for y: 4x + 2y = -6• Subtract 4x• Simplify• Divide both sides by 2• Simplify

- 4x - 4x

2y = -4x - 6

2 2

y = -2x - 3

3) Solve for y: x - 3y = 6

• Subtract x• Simplify• Divide both sides by -3• Simplify

- x - x

-3y = -x + 6

-3 -3

6

3

xy

2

3

xy

or

Graphing Steps

1) Isolate the variable (solve for y).

2) Make a t-table. If the domain is not given, pick your own values.

3) Plot the points on a graph.

4) Connect the points.

Make a t-tableIf f(x) = 2x + 4, complete a table using

the domain {-2, -1, 0, 1, 2}.

2(-2) + 4 = 0 (-2, 0)

2(-1) + 4 = 2 (-1, 2)

2(0) + 4 = 4 (0, 4)

2(1) + 4 = 6 (1, 6)

2(2) + 4 = 8 (2, 8)

x f(x)-2

-1

0

1

2

ordered pair

1) Given the domain {-2, -1, 0, 1, 2},

graph 3x + y = 6

-3(-2) + 6 = 12 (-2, 12)

-3(-1) + 6 = 9 (-1, 9)

-3(0) + 6 = 6 (0, 6)

-3(1) + 6 = 3 (1, 3)

-3(2) + 6 = 0 (2, 0)

x -3x + 6 ordered pair

1. Solve for y: 3x + y = 6

Subtract 3x - 3x - 3x

y = -3x + 62. Make a table

-2

-1

0

1

2

Bonus questions!What is the x-intercept?

(2, 0)What is the y-intercept?

(0, 6)Does the line increase or decrease?

Decrease

1) Given the domain {-2, -1, 0, 1, 2},

graph 3x + y = 63. Plot the points

(-2,12), (-1,9), (0,6), (1,3), (2,0)

4. Connect the points.

1. .

2. .

3. .

4. .

Ex.2) Which is the graph of y = x – 4?

Standard FormAx + By = C

A, B, and C have to be integers

An equation is LINEAR (the graph is a straight line) if it can be written in standard form.

Determine whether each equation is a linear equation.

3) 4x = 7 + 2y

Can you write this in the form

Ax + By = C?

4x - 2y = 7

A = 4, B = -2, C = 7

This is linear!

4) 2x2 - y = 7Can you write it in standard form?

NO - it has an exponent!Not linear

5) x = 12x + 0y = 12

A = 1, B = 0, C = 12Linear

Determine whether each equation is a linear equation.

Here’s the cheat sheet! An equation that is linear does NOT contain the following:

1. Variables in the denominator

2. Variables with exponents

3. Variables multiplied with other variables.

xy = 12

32y

x

2 3y x

Is this equation linear?

1. Yes

2. No

4 3x y

Standard Formx – 4y = 3

Is this equation linear?

1. Yes

2. No

29 4y x

Exponents are not allowed!

Is this equation linear?y = -3

1. Yes

2. No

Standard Form0x + y = -3

x and y -intercepts

●The x-intercept is the point where a line crosses the x-axis.The general form of the x-intercept is (x, 0).

The y-coordinate will always be zero.

●The y-intercept is the point where a line crosses the y-axis.The general form of the y-intercept is (0, y).

The x-coordinate will always be zero.

To find intercepts….

●To find the x-intercept, plug in 0 for y.

●To find the y-intercept, plug in 0 for x.

Find the x and y- interceptsof x = 4y – 5

● x-intercept:

● Plug in y = 0

x = 4y - 5

x = 4(0) - 5

x = 0 - 5

x = -5

● (-5, 0) is the

x-intercept

● y-intercept:

● Plug in x = 0

x = 4y - 5

0 = 4y - 5

5 = 4y

= y

● (0, ) is the

y-intercept

5

4

5

4

Find the x and y-interceptsof g(x) = -3x – 1*

●x-intercept

●Plug in y = 0

g(x) = -3x - 1

0 = -3x - 1

1 = -3x

= x

●( , 0) is the

x-intercept

●y-intercept

●Plug in x = 0

g(x) = -3(0) - 1

g(x) = 0 - 1

g(x) = -1

●(0, -1) is the

y-intercept

*g(x) is the same as y

1

3

1

3

Find the x and y-intercepts of 6x - 3y =-18

●x-intercept

●Plug in y = 0

6x - 3y = -18

6x -3(0) = -18

6x - 0 = -18

6x = -18

x = -3

●(-3, 0) is the

x-intercept

●y-intercept

●Plug in x = 0

6x -3y = -18

6(0) -3y = -18

0 - 3y = -18

-3y = -18

y = 6

●(0, 6) is the

y-intercept

Find the x and y-intercepts of x = 3

● y-intercept

● A vertical line never crosses the y-axis.

● There is no y-intercept.

● x-intercept

● Plug in y = 0.

There is no y. Why?

● x = 3 is a vertical line so x always equals 3.

● (3, 0) is the x-intercept.x

y

Find the x and y-intercepts of y = -2

● x-intercept

● Plug in y = 0.

y cannot = 0 because

y = -2.● y = -2 is a horizontal

line so it never crosses

the x-axis.

●There is no x-intercept.

● y-intercept

● y = -2 is a horizontal line

so y always equals -2.

● (0,-2) is the y-intercept.

x

y

Graphing Equations

●Example: Graph the equation -5x + y = 2Solve for y first.

-5x + y = 2 Add 5x to both sides y = 5x + 2

●The equation y = 5x + 2 is in slope-intercept form, y = mx+b. The y-intercept is 2 and the slope is 5. Graph the line on the coordinate plane.

x

y

Graph y = 5x + 2

Graphing Equations

Graph 4x - 3y = 12

●Solve for y first

4x - 3y =12 Subtract 4x from both sides

-3y = -4x + 12 Divide by -3

y = x + Simplify

y = x – 4

●The equation y = x - 4 is in slope-intercept form, y=mx+b. The y -intercept is -4 and the slope is . Graph the line on the coordinate plane.

Graphing Equations

12-3

43

43

43

-4-3

Graph y = x - 4

x

y

43

Graphing Equations