copyright © 2013, 2009, 2006 pearson education, inc. 1 1 section 3.4 the slope- intercept form of...

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1

Section 3.4

The Slope-Intercept Formof the Equation

of a Line

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1


Objective #1 Find a line’s slope and y-intercept

from its equation.


Slope-Intercept Form

In the form of the equation of the line where the equation is written as y = mx + b, if x = 0, then y = b.

Therefore, b is the y-intercept for the equation of the line. Furthermore, the x coefficient m is the slope of the line.

This form of the equation is called the “Slope- Intercept Form” since we can easily see both of these, the slope and the y-intercept, in the actual equation.


Slopes of Lines

Slope-Intercept Form of the Equation of a Line

y = mx + b with slope m, and y-intercept b


Find the slope and y-intercept of the line y = 2x – 7

The slope is 2.

Slope-Intercept Form


Slopes of Lines

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Give the slope and y-intercept for the line whose equation is:

5 3 7

5 5

y x

5

7

5

3 xy

Subtract 3x from both sides

Simplify

Simplify

Hint: You should solve for y so as to have the equation in slope intercept form.


EXAMPLE EXAMPLE Find the slope and y-intercept of the line 2x + 3y = 9

33

23

9

3

23

92

3

3

923

932

xy

xy

xy

xy

yx

Slope – Intercept Form


Objective #1: Examples

1a. Find the slope and the y-intercept of the line: 2

43

y x

The slope is the x-coefficient, which is 2

.3

m

The y-intercept is the constant term, which is 4.



1a. Find the slope and the y-intercept of the line: 2

43

y x

The slope is the x-coefficient, which is 2

.3

m




1b. Find the slope and the y-intercept of the line: 7 6x y

First, solve the equation for y. 7 6 7 6x y y x

The slope is the x-coefficient, which is 7.m



Objective #2 Graph lines in slope-intercept form.


Slopes of Lines

Graphing y = mx + b Using the Slope and y-Intercept1) Plot the point containing the y-intercept on the y-axis. This is the point

(0,b).

2) Obtain a second point using the slope, m. Write m as a fraction, and use rise over run, starting at the point on the y-axis, to plot this point.

3) Use a straightedge to draw a line through the two points. Draw arrowheads at the ends of the line to show that the line continues indefinitely in both directions


Graphing Lines

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Graph the line whose equation is y = 2x + 3.

The equation y = 2x + 3 is in the form y = mx + b.

Now that we have identified the slope and the y-intercept, we use the three steps in the box to graph the equation.

The slope is 2 and the y-intercept is 3.


Graphing Lines

1) Plot the point containing the y-intercept on the y-axis. The y-intercept is 3. We plot the point (0,3), shown below.

CONTINUEDCONTINUED

(0,3)


Graphing Lines

2) Obtain a second point using the slope, m. Write m as a fraction, and use rise over run, starting at the point containing the y-intercept, to plot this point. We express the slope, 2, as a fraction.

CONTINUEDCONTINUED

(0,3)We plot the second point on the line by starting at (0, 4), the first point. Based on the slope, we move 2 units up (the rise) and 1 unit to the right (the run). This puts us at a second point on the line, (1, 5), shown on the graph.

(1,5)Run

Rise

1

2m


Graphing Lines

3) Use a straightedge to draw a line through the two points. The graph of y = 2x + 3 is show below.

CONTINUEDCONTINUED

(0,3)(1,5)



2a. Graph: 3 2y x

The y-intercept is –2, so plot the point (0, 2).

The slope is 3

3 or .1

m m

Find another point by going up 3 units and to the right 1 unit.

Use a straightedge to draw a line through the two points.



2a. Graph: 3 2y x

The y-intercept is –2, so plot the point (0, 2).

The slope is 3

3 or .1

m m

Find another point by going up 3 units and to the right 1 unit.




2b. Graph: 3

15

y x

The y-intercept is 1, so plot the point (0,1).

The slope is 3

.5

m

Find another point by going up 3 units and to the right 5 units.




2b. Graph: 3

15

y x


The slope is 3

.5

m

Find another point by going up 3 units and to the right 5 units.



Objective #3 Use slope and y-intercept to graph Ax + By = C.


Graph using the Slope and y-Intercept

Earlier in this chapter, we considered linear equations of the form . We used x- and y-intercepts, as well as checkpoints, to graph these equations. It is also possible to obtain graphs by using the slope and y-intercept. To do this, begin by solving for y. This will put the equation in slope-intercept form. Then use the three-step procedure to graph the equation.

Ax By C

Ax By C


3. Graph 3 4 0x y by using slope and y-intercept.

Solve for y: 3 4 0

4 3

34

x y

y x

y x


The slope is 3

.4

m

Objective #3: Example


3. Graph 3 4 0x y by using slope and y-intercept.

Solve for y: 3 4 0

4 3

34

x y

y x

y x


The slope is 3

.4

m




Find another point by going down 3 units and to the right 4 units. Use a straightedge to draw a line through the two points.


Objective #4 Use slope and y-intercept to model data.


Slope as Rate of Change

EXAMPLEEXAMPLE

A linear function that models data is described. Find the slope of each model. Then describe what this means in terms of the rate of change of the dependent variable y per unit change in the independent variable x.

The linear function y = 2x + 24 models the average cost, y in dollars of a retail drug prescription in the United States, x years after 1991.

SOLUTIONSOLUTION

The slope of the linear model is 2. This means that every year (since 1991) the average cost in dollars of a retail drug prescription in the U.S. has increased approximately $2.


Linear Modeling

EXAMPLEEXAMPLE

Find a linear function, in slope-intercept form or a constant function that models the given description. Describe what each variable in your model represents. Then use the model to make a prediction of the number of U.S. Internet users in 2010.

SOLUTIONSOLUTION

In 1998, there were 84 million Internet users in the United States and this number has increased at a rate of 21 million users per year since then.

I(x) = 21x + 84, where I(x) is the number of U.S. Internet users (in millions) x years after 1998. We predict in 2010, there will be I(12) = 21(12) + 84 = 336 million U.S. Internet users.


4. The figure shows the percentage of the U.S. population who had graduated from high school and from college in 1960 and 2010.




4a. Use the two points for college in the figure to find an equation in the form y mx b that models the percentage of college graduates in the U.S. population, y, x years after 1960.

The y-intercept is 8 and the slope is The equation is 0.32 8.y x

Change in 24 8 160.32

Change in 50 0 50y

mx



4b. Use the model from part (a) to project the percentage of college graduates in 2020.

0.32 8

0.32(60) 8

27.2

y x

The model projects that 27.2% of the U.S. population will be college graduates in 2020.

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