Download 4 14.00 7 21.50 18 Express the cost as a linear function of the

2-4
2-4 Writing
Writing Linear
Linear Functions
Functions
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra
Holt
Algebra
2 2
2-4
Writing Linear Functions
Warm Up
Write each function in slope-intercept form.
1. 4x + y = 8
y = –4x + 8
2. –y = 3x
y = –3x
3. 2y = 10 – 6x
y = –3x + 5
Determine whether each line is vertical or
horizontal.
3
4. x =
4
vertical
Holt Algebra 2
5. y = 0
horizontal
2-4
Writing Linear Functions
Objectives
Use slope-intercept form and pointslope form to write linear functions.
Write linear functions to solve
problems.
Holt Algebra 2
2-4
Writing Linear Functions
Vocabulary
Point-slope form
Holt Algebra 2
2-4
Writing Linear Functions
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.
Holt Algebra 2
2-4
Writing Linear Functions
Example 1: Writing the Slope-Intercept Form of the
Equation of a Line
Write the equation of the graphed line in slopeintercept form.
Step 1
Holt Algebra 2
Identify the y-intercept.
The y-intercept b is 1.
2-4
Writing Linear Functions
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.
3
4
–4
–3
Slope is rise = –3 = – 3 .
run
Holt Algebra 2
4
4
2-4
Writing Linear Functions
Example 1 Continued
Step 3
Write the equation in slope-intercept form.
y = mx + b
3
y=– x+1
4
m= –
3
and b = 1.
4
The equation of the line is y = –
Holt Algebra 2
3
x + 1.
4
2-4
Writing Linear Functions
Check It Out! Example 1
Write the equation of the graphed line in slopeintercept form.
Step 1 Identify the y-intercept.
The y-intercept b is 3.
Holt Algebra 2
2-4
Writing Linear Functions
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
Slope is rise = 3 .
run
Holt Algebra 2
4
3
3
4
3
4
2-4
Writing Linear Functions
Check It Out! Example 1 Continued
Step 3
Write the equation in slope-intercept form.
y = mx + b
y=
3
x+3
4
m=
3
and b = 3.
4
The equation of the line is y =
Holt Algebra 2
3
x + 3.
4
2-4
Writing Linear Functions
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.
Holt Algebra 2
2-4
Writing Linear Functions
Helpful Hint
If you reverse the order of the points in Example
2B, the slope is still the same.
m=
Holt Algebra 2
6 – 16
5 – 11
=
– 10
–6
=
5
3
2-4
Writing Linear Functions
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.
Holt Algebra 2
2-4
Writing Linear Functions
Example 2B: Finding the Slope of a Line Given Two or
More Points
Find the slope of the line.
x
4
8
12
16
y
2
5
8
11
Choose any
Let (x1, y1) be (4, 2) and (x2, y2) be (8, 5). two points.
Use the slope formula.
The slope of the line is 3 .
4
Holt Algebra 2
2-4
Writing Linear Functions
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.
Holt Algebra 2
2-4
Writing Linear Functions
Check It Out! Example 2A
Find the slope of the line.
Let (x1, y1) be (–4, –1)
and (x2, y2) be (–2, 1).
x
y
–6
–3
–4
–1
–2
1
Choose any two points.
Use the slope formula.
The slope of the line is 1.
Holt Algebra 2
2-4
Writing Linear Functions
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.
Holt Algebra 2
2-4
Writing Linear Functions
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.
Holt Algebra 2
2-4
Writing Linear Functions
Example 3: Writing Equations of Lines
In slope-intercept form, write the equation
of the line that contains the points in the
table.
x
–8
–4
y
–5
–3.5
4
–0.5
8
1
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.
Holt Algebra 2
2-4
Writing Linear Functions
Example 3 Continued
Method A Point-Slope Form
Using (8, 1):
Rewrite in slopeintercept form.
y – y1 = m(x – x1)
Distribute.
Substitute.
Simplify.
Holt Algebra 2
Solve for y.
2-4
Writing Linear Functions
Example 3 Continued
Method B Slope-intercept Form
Using (8, 1), solve for b.
y = mx + b
Rewrite the equation
using m and b.
y = mx + b
Substitute.
1=3+b
Simplify.
b = –2
Solve for b.
The equation of the line is
Holt Algebra 2
.
2-4
Writing Linear Functions
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)
y – (3) = –5(x – 1)
y – 3 = –5(x – 1)
Substitute.
Simplify.
Rewrite in slope-intercept form.
y – 3 = –5(x – 1)
y – 3 = –5x + 5
y = –5x + 8
Holt Algebra 2
Distribute.
Solve for y.
The equation of the
slope is y = –5x + 8.
2-4
Writing Linear Functions
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
5 = (2)2 + b
using m and b.
5=4+b
y = mx + b
y = 2x + 1
1=b
The equation of the line is y = 2x + 1.
Holt Algebra 2
2-4
Writing Linear Functions
Example 4A: Entertainment Application
The table shows the rents and selling prices
of properties from a game.
Selling Price
Rent
($)
($)
Express the rent as a function
75
9
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).
Holt Algebra 2
90
12
160
26
250
44
2-4
Writing Linear Functions
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.
Holt Algebra 2
2-4
Writing Linear Functions
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 $40.
Holt Algebra 2
2-4
Writing Linear Functions
Check It Out! Example 4a
Express the cost as a linear
function of the number of
items.
Let x = items and y = cost.
Find the slope by choosing two
points. Let (x1, y1) be (4, 14)
and (x2, y2) be (7, 21.50).
Holt Algebra 2
Items
Cost ($)
4
14.00
7
21.50
18
2-4
Writing Linear Functions
Check It Out! Example 4a Continued
To find the equation for the number of items,
use point-slope form.
y – y1 = m(x – x1)
y – 14 = 2.5(x – 4)
y = 2.5x + 4
Holt Algebra 2
Use the data in the first row
of the table.
Simplify.
2-4
Writing Linear Functions
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.
y = 2.5(18) + 4
Substitute.
y = 45 + 4
y = 49
The cost for 18 items is $49.
Holt Algebra 2
2-4
Writing Linear Functions
By comparing slopes, you can determine if the
lines are parallel or perpendicular. You can also
write equations of lines that meet certain
criteria.
Holt Algebra 2
2-4
Writing Linear Functions
Holt Algebra 2
2-4
Writing Linear Functions
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)
m = 1.8
Parallel lines have equal slopes.
Use y – y1 = m(x – x1) with (x1, y1) =
y – 2 = 1.8(x – 5)
(5, 2).
y – 2 = 1.8x – 9
y = 1.8x – 7
Holt Algebra 2
Distributive property.
Simplify.
2-4
Writing Linear Functions
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)
The slope of the given line is
, so the slope of
the perpendicular line is the opposite reciprocal,
Use y – y1 = m(x – x1). y + 2 is
equivalent to y – (–2).
Distributive property.
Simplify.
Holt Algebra 2
.
2-4
Writing Linear Functions
Check It Out! Example 5a
Write the equation of the line in slope-intercept
form.
parallel to y = 5x – 3 and through (1, 4)
m=5
y – 4 = 5(x – 1)
y – 4 = 5x – 5
y = 5x – 1
Holt Algebra 2
Parallel lines have equal slopes.
Use y – y1 = m(x – x1) with
(x1, y1) = (5, 2).
Distributive property.
Simplify.
2-4
Writing Linear Functions
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)
The slope of the given line is
, so the slope of
the perpendicular line is the opposite reciprocal,
Use y – y1 = m(x – x1). y + 2 is
equivalent to y – (–2).
Distributive property.
Simplify.
Holt Algebra 2
.
2-4
Writing Linear Functions
Check It Out! Example 5b
Write the equation of the line in slope-intercept
form.
perpendicular
to and through (0, –2)
The slope of the given line is
, so the slope of
the perpendicular, line is the opposite reciprocal
Use y – y1 = m(x – x1). y + 2
is equivalent to y – (–2).
Distributive property.
Simplify.
Holt Algebra 2
.
2-4
Writing Linear Functions
Lesson Quiz: Part I
Write the equation of each line in slopeintercept form.
1.
y = –2x –1
2. parallel to y = 0.5x + 2 and through (6, 1)
y = 0.5x – 2
3. perpendicular to
Holt Algebra 2
and through (4, 4)
2-4
Writing Linear Functions
Lesson Quiz: Part II
4. Express the catering cost as a function of
the number of people. Find the cost of
catering a meal for 24 people.
Number in Group
Cost ($)
4
7
15
98
134
230
f(x) = 12x + 50; $338
Holt Algebra 2