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Objectives

• Find the slopes of lines• Write and graph linear equations• Model data with linear functions and make

predictions

Linear Functions and Slopes

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0011 0010 1010 1101 0001 0100 1011VOCABULARY

Linear equationSlopePoint-slope formSlope-intercept form

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x

y

x2 – x1

y2 – y1

change in y

change in x

The slope of the line passing through the two points (x1, y1) and (x2, y2) is given by the formula

(x1, y1)

(x2, y2)

Slope Formula

(where

The slope of a line is a number, m, which measures its steepness.

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Example: Find the slope of the line passing through the points (2, 3) and (4, 5).

y2 – y1

x2 – x1

m = 5 – 3

4 – 2= =

22

= 1

2

2(2, 3)

(4, 5)

x

y

x1 x2y1 y2

Use the slope formula.

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Your Turn

Find the slope of the line passing through each pair of points.

( and (2, 8)

(4, 5) and (8, 4)

2

14

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Slope of Horizontal Lines

• Slope of a horizontal line is 0

• Equation of a horizontal line that passes through the point (a,b):

(0,4)

x

y

0

(1,4) 4 4 00

1 0 1m

y b

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Slope of Vertical Lines

• Slope of a vertical line is undefined

• Equation of a vertical line that passes through the point (a,b):

undefinedm

04

4404

ax

x

y

0

(4,4)

4 (4,0)

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𝒙=𝟒

𝒚=−𝟐

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Because the slope of line is constant, it is possible to use any point on a line and the slope of the line to write an equation of the line in point-slope form.

All you need is a point and the slope.

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Point-Slope FormThe Point-Slope form is derived from the slope formula.

2 1

2 1

y ym

x x

1

1

y ym

x x

11 1

1

y yx x m x x

x x

1 1 1 1 or x x m y y y y m x x

Slope Formula

Change y2, x2 to just y and x.

Multiple both sides by the denominator.

Point-Slope Form

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0011 0010 1010 1101 0001 0100 1011A linear equation written in the form y – y1 = m(x – x1) is in point-slope form.

The graph of this equation is a line with slope m passing through the point (x1, y1).

Example:

The graph of the equation

y – 3 = - (x – 4) is a line

of slope m = - passing

through the point (4, 3).

1

2 1

2

(4, 3)

m = -1

2

x

y

4

4

8

8

Point-Slope Form

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Write an equation in point-slope form for the line with slope 4 that passes through the point . Then solve the equation for .

Use the point-slope form of the equation.

𝑦− 𝑦1=𝑚 (𝑥−𝑥2)

Substitute the given values

Point-slope form

Solve for

Distribution property

Combine like terms

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Write an equation in point-slope form for the line passing through the points and . Then solve the equation for .

Rule # 1: We need to find the slope, whether it is given to us or it needs to be calculated.

y2 – y1

x2 – x1

m =

𝑥1 𝑥2𝑦 1 𝑦 2

Substitute the slope, , and either coordinate into the point-slope formula .

𝑦− (− 3 )=(−32 ) (𝑥− 4 )

To be continued

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𝑦− (− 3 )=(−32 ) (𝑥− 4 )

Point-slope form

𝑦=−32𝑥+6 −3

Final answer

The above answer is the slope-intercept form of the equation.

where the slope and is the y-intercept.

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

To graph an equation in slope-intercept form:

1. Write the equation in the form y = mx + b. Identify m and b.

The slope is m and the y-intercept is b.

2. Plot the y-intercept (0, b).

3. Starting at the y-intercept, find another point on the line using the slope.4. Draw the line through (0, b) and the point located using the slope.

Slope-Intercept Form

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1

Example: Graph the line y = 2x – 4.

2. Plot the y-intercept, (0, - 4).

1. The equation y = 2x – 4 is in the slope-intercept form. So, m = 2 and b = - 4.

3. The slope is 2.

The point (1, -2) is also on the line.

1= change in y

change in xm = 2

4. Start at the point (0, 4). Count 1 unit to the right and 2 units up to locate a second point on the line.

2

x

y

5. Draw the line through (0, 4) and (1, -2).

(0, - 4)

(1, -2)

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(0 ,5)

(1 , 2)

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(2 , −2)(0 ,− 3)

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General Form of the Equation of a Line

Every line has an equation that can be written in the general form

where and are real numbers and and are not both zero.

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Find the slope and the y-intercept of the line whose equation is .

We need to change the equation from general form to slope-intercept form.

2 𝑦=−3 𝑥+4

𝑦=−32𝑥+2

The slope is The y-intercept is

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Using Intercepts to Graph

Find the x-intercept. Let and solve for . Plot the point on the x-axis.

Find the y-intercept. Let and solve for . Plot the point on the y-axis.

Draw a line through the two points, using arrowheads on the ends to indicate the line continues in both directions indefinitely.

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The point (0,4) is the y-intercept.

The point (6,0) is the x-intercept.

x

y

2-2

This is the graph of the equation .

(0,4)

(6,0)

Linear Equations

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Summary of Equations of Lines

1. General form:

2. Vertical line:

3. Horizontal line:

4. Slope-intercept form:

5. Point-slope form:

0Ax By C

x ay b

y mx b

1 1y y m x x

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Linear Model

Writing an equation of a line that models real data: If the data changes at a fairly constant rate, the rate of change is the slope. An initial condition would be the y-intercept.

• Example: Suppose there is a flat rate of $.20 plus a charge of $.10/minute to make a phone call. Write an equation that gives the cost y for a call of x minutes. Note: The initial condition is the flat rate of $.20 and the rate of change is $.10/minute.

Solution: y = .10x + .20

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Linear Model

Writing an equation of a line that models real data: If the data changes at a fairly constant rate, the rate of change is the slope. An initial condition would be the y-intercept.

• Example: The percentage of mothers of children under 1 year old who participated in the US labor force is shown in the table. Find an equation that models the data.

Using (1980,38) and (1998,59)Year Percent

1980 381984 471988 511992 541998 59

59 38 21

1998 1980

1.1

181.1

67

67

38

m

m

y x

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Your Turn

The net sales for a car manufacturer were $14.61 billion in 2005 and $15.78 billion in 2006. Write a linear equation giving the net sales y in terms of x, where x is the number of years since 2000. Then use the equation to predict the net sales for 2007.

Answer: y=1.17x+8.76, predicted sales for 2007 is $16.95 billion.

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Summary of Linear GraphsForms of Linear Equations

Equation Description When to Use

Y = mx + bSlope-Intercept Form

slope is m y-intercept is (0,b)

Given an equation, the slope and y-intercept can be easily identified and used to graph

y - y1 = m(x-x1)Point-Slope Form

slope is m line passes through (x1,y1)

This form is ideal to use when given the slope of a line and one point on the line or given

two points on the line.

Ax+By+C=0

Standard Form (A,B, and C are integers, A>0)

Slope is -(A/B) x-intercept is (C/A,0) y-intercept is (0,C/B)

X- and y-intercepts can be found quickly

y = b

Horizontal line slope is 0

y-intercept is (0,b)Graph intersects only the y axis, is parallel to the x-axis

x = a

Vertical line slope is undefined x-intercept is (a,0)

Graph intersects only the x axis, is parallel to the y-axis

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