Use slopeintercept form and pointslope form to write linear functions'

1
Objectives
Use slope-intercept form and point-slope form to
write linear functions. Write linear functions to
solve problems.
2
Vocabulary
Point-slope form
3
Recall from Lesson 2-3 that the slope-intercept
form of a linear equation is y mx b, where m
is the slope of the line and b is its
y-intercept.
In Lesson 2-3, you graphed lines when you were
given the slope and y-intercept. In this lesson
you will write linear functions when you are
given graphs of lines or problems that can be
modeled with a linear function.
4
Example 1 Writing the Slope-Intercept Form of
the Equation of a Line
Write the equation of the graphed line in
slope-intercept form.
Identify the y-intercept. The y-intercept b is 1.
Step 1
5
Example 1 Continued
Step 2 Find the slope.
Choose any two convenient points on the line,
such as (0, 1) and (4, 2). Count from (0, 1) to
(4, 2) to find the rise and the run. The rise is
3 units and the run is 4 units.
6
Example 1 Continued
Step 3
Write the equation in slope-intercept form.
y mx b
7
Check It Out! Example 1
Write the equation of the graphed line in
slope-intercept form.
Identify the y-intercept. The y-intercept b is 3.
Step 1
8
Check It Out! Example 1 Continued
Step 2 Find the slope.
Choose any two convenient points on the line,
such as (4, 0) and (0, 3). Count from (4, 0) to
(0, 3) to find the rise and the run. The rise is
3 units and the run is 4 units
3
9
Check It Out! Example 1 Continued
Step 3
Write the equation in slope-intercept form.
y mx b
10
Notice that for two points on a line, the rise is
the differences in the y-coordinates, and the run
is the differences in the x-coordinates. Using
this information, we can define the slope of a
line by using a formula.
11
(No Transcript)
12
Example 2A Finding the Slope of a Line Given Two
or More Points
Find the slope of the line through (1, 1) and
(2, 5).
Let (x1, y1) be (1, 1) and (x2, y2) be (2, 5).
Use the slope formula.
The slope of the line is 2.
13
Example 2B Finding the Slope of a Line Given Two
or More Points
Find the slope of the line.
Choose any two points.
Let (x1, y1) be (4, 2) and (x2, y2) be (8, 5).
Use the slope formula.
14
Example 2C Finding the Slope of a Line Given Two
or More Points
Find the slope of the line shown.
Let (x1, y1) be (0,2) and (x2, y2) be (1, 2).
The slope of the line is 0.
15
Check It Out! Example 2A
Find the slope of the line.
Let (x1, y1) be (4, 1) and (x2, y2) be (2, 1).
Choose any two points.
Use the slope formula.
The slope of the line is 1.
16
Check It Out! Example 2B
Find the slope of the line through (2,5) and
(3, 5).
Let (x1, y1) be (2, 5) and (x2, y2) be (3, 5).
Use the slope formula.
The slope of the line is 0.
17
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.
18
Example 3 Writing Equations of Lines
In slope-intercept form, write the equation of
the line that contains the points in the table.
First, find the slope. Let (x1, y1) be (8, 5)
and (x2, y2) be (8, 1).
Next, choose a point, and use either form of the
equation of a line.
19
Example 3 Continued
Method A Point-Slope Form
Rewrite in slope-intercept form.
Using (8, 1)
y y1 m(x x1)
Distribute.
Substitute.
Solve for y.
Simplify.
20
Example 3 Continued
Method B Slope-intercept Form
Using (8, 1), solve for b.
Rewrite the equation using m and b.
y mx b
y mx b
Substitute.
Simplify.
1 3 b
Solve for b.
b 2
21
Check It Out! Example 3a
Write the equation of the line in slope-intercept
form with slope 5 through (1, 3).
Method A Point-Slope Form
y y1 m(x x1)
Substitute.
y (3) 5(x 1)
y 3 5(x 1)
Simplify.
Rewrite in slope-intercept form.
y 3 5(x 1)
y 3 5x 5
Distribute.
The equation of the slope is y 5x 8.
Solve for y.
y 5x 8
22
Check It Out! Example 3b
Write the equation of the line in slope-intercept
form through (2, 3) and (2, 5).
First, find the slope. Let (x1, y1) be (2,3)
and (x2, y2) be (2, 5).
Method B Slope-Intercept Form
y mx b
Rewrite the equation using m and b.
5 (2)2 b
5 4 b
y mx b
y 2x 1
1 b
The equation of the line is y 2x 1.
23
Example 4A Entertainment Application
The table shows the rents and selling prices of
properties from a game.
Express the rent as a function of the selling
price. Let x selling price and y rent. Find
the slope by choosing two points. Let (x1, y1) be
(75, 9) and (x2, y2) be (90, 12).
24
Example 4A Continued
To find the equation for the rent function, use
point-slope form.
y y1 m(x x1)
Use the data in the first row of the table.
Simplify.
25
Example 4B Entertainment Application
Graph the relationship between the selling price
and the rent. How much is the rent for a property
with a selling price of 230?
To find the rent for a property, use the graph or
substitute its selling price of 230 into the
function.
Substitute.
y 46 6
y 40
The rent for the property is 40.
26
Check It Out! Example 4a
Express the cost as a linear function of the
number of items. Let x items and y
cost. Find the slope by choosing two points.
Let (x1, y1) be (4, 14) and (x2, y2) be (7,
21.50).
27
Check It Out! Example 4a Continued
To find the equation for the number of items, use
point-slope form.
y y1 m(x x1)
Use the data in the first row of the table.
y 14 2.5(x 4)
Simplify.
y 2.5x 4
28
Check It Out! Example 4b
Graph the relationship between the number of
items and the cost. Find the cost of 18 items.
To find the cost, use the graph or substitute the
number of items into the function.
Substitute.
y 2.5(18) 4
y 45 4
y 49
The cost for 18 items is 49.
29
By comparing slopes, you can determine if the
lines are parallel or perpendicular. You can also
write equations of lines that meet certain
criteria.
30
(No Transcript)
31
Example 5A Writing Equations of Parallel and
Perpendicular Lines
Write the equation of the line in slope-intercept
form.
parallel to y 1.8x 3 and through (5, 2)
Parallel lines have equal slopes.
m 1.8
Use y y1 m(x x1) with (x1, y1) (5, 2).
y 2 1.8(x 5)
Distributive property.
y 2 1.8x 9
Simplify.
y 1.8x 7
32
Example 5B Writing Equations of Parallel and
Perpendicular Lines
Write the equation of the line in slope-intercept
form. perpendicular to and
through (9, 2)
Use y y1 m(x x1). y 2 is equivalent to y
(2).
Distributive property.
Simplify.
33
Check It Out! Example 5a
Write the equation of the line in slope-intercept
form.
parallel to y 5x 3 and through (1, 4)
Parallel lines have equal slopes.
m 5
Use y y1 m(x x1) with (x1, y1) (5, 2).
y 4 5(x 1)
y 4 5x 5
Distributive property.
Simplify.
y 5x 1
34
Example 5B Writing Equations of Parallel and
Perpendicular Lines
Write the equation of the line in slope-intercept
form. perpendicular to and
through (9, 2)
Use y y1 m(x x1). y 2 is equivalent to y
(2).
Distributive property.
Simplify.
35
Check It Out! Example 5b
Write the equation of the line in slope-intercept
form. perpendicular to and
through (0, 2)
Use y y1 m(x x1). y 2 is equivalent to y
(2).
Distributive property.
Simplify.