Chapter 1 Linear Functions. Slopes and Equations of Lines The Rectangular Coordinate System – The...

Chapter 1

Linear Functions

Slopes and Equations of Lines

The Rectangular Coordinate System– The horizontal number line is the x-axis– The vertical number line is the y-axis– The point where the axes intersect is the origin

(the 0 point)


The point given as (2,3) is also called an ordered pair.– Where 2 is the x coordinate, the first coordinate,

the abscissa, or the horizontal distance– Where 3 is the y coordinate, the second

coordinate, the ordinate, or the vertical distance


Intercepts of a Line– The y-intercept of a line is the point (0,b) where

the line intersects the y axis. To find b, substitute 0 for x in the equation of the line and solve for y.

– The x-intercept of a line is the point (a,0) where the line intersects the x-axis. To find a, substitute 0 for y in the equation of a line and solve for x.


Slope– Slope (m) is a measure of the slant of a line.– The slope of the line between two points is the

ratio of the change in y to the change in x.

x

y


– The slope of a line is defined as the vertical change (the “rise”) over the horizontal change (the “run”) as one travels along the line.

12

12

xx

yym


Slope of a Line

Slopes and Equations of Lines Point-Slope Form of the Equation of a Line– Recall the slope-intercept from of a line is

Y = mx + b– where m is slope and b is the y-intercept

CASE A GIVEN A POINT AND THE SLOPE

– The equation of the line passing through P(x1,y1) and with slope m is

y – y1 = m( x – x1 ) (the point-slope formula)

So given a slope and a point (other than y-intercept) – We replace m with the given slope– We replace x1 and y1 with the point coordinates– Simplify– Write the equation in the form of y = mx + b


CASE B

GIVEN TWO POINTS ONLY

– The equation of the line passing through P(x1,y1) and P(x2,y2) (slope is not given)

We first use the slope formula to determine slope Then we use one of the given points and the slope in the point-

slope formula

y – y1 = m( x – x1 )


CASE C

GIVEN A POINT P(x1,y1) AND AN UNDEFINED SLOPE

– Recall that a line with an undefined slope is a vertical line with x = a, where a is the x-coordinate of the x-intercept (x1, 0)

– Recall that for a vertical line, the x-coordinate doesn’t change.

the equation would simply be x = x1


CASE D

GIVEN A POINT P(x1,y1) AND A ZERO SLOPE

– Recall that a line with a zero slope is a horizontal line with y = b, where b is the y-coordinate of the y-intercept (0, b)

– Recall that for a horizontal line, the y-coordinate doesn’t change.

the equation would simply be y = y1


CASE E

GIVEN THE Y-INTERCEPT (0,b) AND SLOPE

– We use the slope-intercept form y = mx +b

– Replace m with the given slope– Replace b with the given y-coordinate


Slopes of Horizontal and Vertical Lines– horizontal lines (y = b) have a slope of 0– vertical lines (x = a) have no defined slope

Slopes of Parallel Lines – Two nonvertical lines are parallel if they have the same slope (they

do not intersect) (i.e., line A has slope of 1/2 and line B has a slope of 1/2)

Slopes of Perpendicular Lines– Two nonvertical lines are perpendicular if their slopes are negative

reciprocals of each other. These lines intersect at right angles (i.e., line A has slope of 1/2 and line B has a slope of -2)

Slopes and Equations of Lines Finding equations of parallel and perpendicular

lines To find equations of parallel and perpendicular lines we

always need to know slope. Recall that

– parallel lines () – two lines that do not intersect - have the same slope.

– perpendicular lines ( ) – two lines that intersect at right angles – have slopes that are negative reciprocals of each other.


Example– Find the equation of a line containing the point

(3,2) and to the line 3x + y = -3. To be parallel, the lines must have the same slope. The

line 3x + y = -3 has a slope of -3 because when we solve for y : y = -3x -3

Given a point (3,2) and a slope -3 we use the point-slope formula

– y – y1 = m( x – x1 ) The equation is y = -3x + 11

Slopes and Equations of Lines Example

– Find the equation of a line containing the point (-1,-3) and to the line 3x - 5y = 2.

To be perpendicular, the lines must have slopes that are negative reciprocals of each other. The line 3x - 5y = 2 has a slope of 3/5 because when we solve for y :

– y = 3/5x -2/5 The slope of the other line has to be -5/3 Given a point (-1,-3) and a slope -5/3 we use the point-slope formula

– y – y1 = m( x – x1 ) The equation is y = -5/3x – 14/3

Slopes and Equations of Lines Graphing Lines– To graph an equation in two variables such as

y = x – 1 when x = -2, -1, 0, 1, 2. Using Substitution

– Substitute each value of x into the equation.– Solve for y– Graph the ordered pairs

Using slope and y-intercept– Place a point at the y-intercept– Starting from the y-intercept, “rise” then “run” based on slope

Linear Functions And Applications

Linear Functions– Notation – f(x)

Note: f, g, or h are often used to name functions

– The function f(x) is defined by y = f(x) = mx + b


Linear FunctionsOperations

The sum of f and g, denoted as f + g, is defined by (f + g)(x) = f(x) + g(x)The difference of f and g, denoted as f – g, is defined by (f – g)(x) = f(x) – g(x)The product of f and g, denoted as f∙g, is defined by (f∙g)(x) = f(x)g(x)The quotient of f and g, denoted as f/g, is defined by (f/g)(x) = where g(x)0

)(

)(

xg

xf


Example– Let f(x) = 4x – 1. Find f(-2), f(-1), f(0), f(1), f(2)

f(-2) = 4(-2) – 1 = -8 – 1 = -9

f(-1) = 4(-1) – 1 - -4 – 1 = -5

f(0) = 4(0) – 1 = 0 -1 = -1

f(1) = 4(1) – 1 = 4 – 1 = 3

f(2) = 4(2) – 1 = 8 – 1 = 7


Supply and Demand– Supply and Demand functions are not necessarily

linear Most approximately linear

– Equilibrium Price/Quantity Equilibrium price of a commodity is the price found at the

point where the supply and demand are equal (i.e., where the supply and demand graphs for that commodity intersect).

Equilibrium quantity is the quantity where the prices from both demand and the supply are equal.


Cost Analysis– The cost of manufacturing an item commonly

consists of two parts:

Fixed cost – designing the product, setting up a factory, training workers, etc.

Marginal cost – approximates the cost of producing one additional item.


Cost Analysis– The Cost Function

C(x) = mx + b

where m represents marginal cost per item

where b represents the fixed cost


Break-Even Analysis– Involves both revenue and profit.– Revenue R(x) = px

where p represents the price per unit of productwhere x represents the number of units sold (demand)

– Profit P(x) is the difference between revenue and cost: P(x) = R(x) – C(x)

Break-even quantity – the number of units at which revenue just equals cost

Break-even point – the point where revenue equals cost.

The Least Squares Line

In practice, how are equations of supply and demand functions found?– Data is collected– Data is plotted

Data lie perfectly along a line – equation easily found using any two points

Data scattered and no line that goes through all points – must find a line that approximates the linear trend of the data as closely as possible (Method of Least Squares)

The Least Squares Line How do we know when we have the best line (least squares line) ? (see Figure 16, p. 30)– The line whose sum of squared vertical distances

from the data points is as small as possible. The least squares line Y = mx + b, where

and

or

22 )()(

))(()(

xxn

yxxynm

22

2

)()(

))(())((

xxn

xyxyxb

n

xmyb

)(


Calculating by hand

– Given “n” pairs of data, find the required sums.x y xy x2

y2

_________________________________________________x y xy x2

y2

– Plug and Chug– Write least squares equation in the form of Y = mx + b.


Coefficient of Correlation– Measures of “goodness of fit” of the least squares

line

2222 )()()()(

))(()(

yynxxn

yxxynr


– The coefficient of correlation (r) is always equal to or between 1 and -1.

– Coefficient of correlation (r) values of exactly 1 or -1 mean that the data points lie exactly on the least squares line.

– If r = 1, the least squares line has a positive slope– If r = -1, the least squares line has a negative slope– If r = 0, there is no linear correlation between the data

points (some nonlinear function might provide a better fit)– The closer the value of r is to 1 or -1, the stronger the linear

relationship between the data points.

Chapter 1 Linear Functions. Slopes and Equations of Lines The Rectangular Coordinate System – The...

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