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12-5 Linear Functions
Check 12-4 Homework
Pre-Algebra
12-5 Linear Functions
Pre-Algebra HOMEWORK
Page 636-637
#8-10 & 21-22
Pre-Algebra
12-5 Linear Functions
Students will be able to solve sequences and represent
functions by completing the following assignments.
•
•
•
•
Learn to find terms in an arithmetic sequence.
Learn to find terms in a geometric sequence.
Learn to find patterns in sequences.
Learn to represent functions with tables, graphs, or
equations.
• Learn to identify linear functions.
• Learn to recognize inverse variation by graphing
tables of data.
Pre-Algebra
12-5 Linear Functions
Today’s Learning Goal Assignment
Learn to
identify linear
functions.
Pre-Algebra
12-5
12-5Linear
LinearFunctions
Functions
Warm Up
Problem of the Day
Lesson Presentation
Pre-Algebra
Pre-Algebra
12-5 Linear Functions
Warm Up
Determine if each relationship represents
a function.
1.
yes
2. y = 3x2 – 1 yes
3. For the function f(x) = x2 + 2, find f(0), f(3),
and f(–2).
Pre-Algebra
2, 11, 6
12-5 Linear Functions
Problem of the Day
Take the first 20 terms of the geometric
sequence 1, 2, 4, 8, 16, 32, . . . .Why
can’t you put those 20 numbers into
two groups such that each group has
the same sum?
All the numbers except 1 are even, so
the sum of the 20 numbers is odd and
cannot be divided into two equal integer
sums.
Pre-Algebra
12-5 Linear Functions
Vocabulary
linear function
Pre-Algebra
12-5 Linear Functions
The graph of a linear function is a line.
The linear function f(x) = mx + b has a
slope of m and a y-intercept of b. You
can use the equation f(x) = mx + b to
write the equation of a linear function
from a graph or table.
Pre-Algebra
12-5 Linear Functions
Additional Example 1: Writing the Equation for a
Linear Function from a Graph
Write the rule for the linear function.
Use the equation f(x) = mx + b.
To find b, identify the y-intercept
from the graph.
b=2
f(x) = mx + 2
Locate another point on the
graph, such as (1, 4).
Substitute the x- and y-values
of the point into the equation,
and solve for m.
Pre-Algebra
12-5 Linear Functions
Additional Example 1 Continued
f(x) =
4=
4=
–2
2=
mx + 2
m(1) + 2
m+2
–2
m
The rule is f(x) = 2x + 2.
Pre-Algebra
(x, y) = (1, 4)
12-5 Linear Functions
Try This: Example 1
Write the rule for the linear function.
y
Use the equation f(x) = mx + b.
To find b, identify the y-intercept
from the graph.
4
2
x
-4
-2
2
-2
-4
Pre-Algebra
4
b=1
f(x) = mx + 1
Locate another point on the
graph, such as (5, 2).
Substitute the x- and y-values
of the point into the equation,
and solve for m.
12-5 Linear Functions
Try This: Example 1 Continued
f(x) = mx + 1
2 = m(5) + 1
2 = 5m + 1
–1
–1
1 = 5m
m=1
5
The rule is f(x) = 1 x + 1.
5
Pre-Algebra
(x, y) = (5, 2)
12-5 Linear Functions
Additional Example 2A: Writing the Equation for a
Linear Function from a Table
Write the rule for the linear function.
A.
x
–2
y
5
–1
0
1
3
1
–1
The y-intercept can be identified
from the table as b = f(0) = 1.
Substitute the x- and y-values of
the point (1, –1) into the equation
f(x) = mx + 1, and solve for m.
f(x) = mx + 1
–1 = m(1) + 1
–1 = m + 1
–1
–1
–2 = m
Pre-Algebra
The rule is
f(x) = –2x + 1.
12-5 Linear Functions
Try This: Example 2A
Write the rule for the linear function.
A.
x
0
y
0
–1 1
1 –1
2 –2
The y-intercept can be identified
from the table as b = f(0) = 0.
Substitute the x- and y-values of
the point (1, –1) into the equation
f(x) = mx + 0, and solve for m.
f(x) = mx + 0
–1 = m(1) + 0
–1 = m
The rule is f(x) = –x.
Pre-Algebra
12-5 Linear Functions
Additional Example 2B: Writing the Equation for a
Linear Function from a Table
Write the rule for the linear function.
B. x
–3
–1
1
3
Pre-Algebra
y
–8
Use two points, such as (1, 4) and
(3, 10), to find the slope.
–2
4
y2 – y1 10 - 4 6
m = x2 – x1 = 3 - 1 = 2 = 3
10
Substitute the x- and y-values of
the point (1, 4) into f(x) = 3x + b,
and solve for b.
12-5 Linear Functions
Additional Example 2B Continued
f(x) = 3x + b
4 = 3(1) + b
4=3+b
(x, y) = (1, 4)
–3 –3
1=b
The rule is f(x) = 3x + 1.
Pre-Algebra
12-5 Linear Functions
Try This: Example 2B
Write the rule for each linear function.
B.
x
0
y
5
1
2
–1
6
7
4
Pre-Algebra
Use two points, such as (0, 5) and
(1, 6), to find the slope.
y2 – y1 6 – 5
m = x2 – x1 = 1 – 0 = 1
1 =1
Substitute the x- and y-values of
the point (0, 5) into f(x) = 1x + b,
and solve for b.
12-5 Linear Functions
Try This: Example 2 Continued
f(x) = mx + b
5 = 1(0) + b
5= b
(x, y) = (0, 5)
The rule is f(x) = x + 5.
Pre-Algebra
12-5 Linear Functions
Example 3: Money Application
A video club cost $15 to join. Each video that
is rented costs $1.50. Find a rule for the
linear function that describes the total cost
of renting videos as a member of the club,
and find the total cost of renting 12 videos.
f(x) = mx + 15
The y-intercept is the cost to join, $15.
16.5 = m(1) + 15 With 1 rental the cost will be $16.50.
16.5 = m + 15
The rule for the function is f(x) =
–15
– 15
1.5x + 15. After 12 video rentals, the
1.5 = m
Pre-Algebra
cost will be f(12) = 1.5(12) + 15 =
18 + 15 = $33.
12-5 Linear Functions
Try This: Example 3
A book club has a membership fee of $20. Each
book purchased costs $2. Find a rule for the linear
function that describes the total cost of buying
books as a member of the club, and find the total
cost of buying 10 books.
f(x) = mx + 20
The y-intercept is the cost to join, $20.
With 1 book purchase the cost will be
22 = m(1) + 20
$22.
22 = m + 20
The rule for the function is
–20
– 20
f(x) = 2x + 20. After 10 book
2=m
purchases, the cost will be
f(10) = 2(10) + 20 = 20 + 20 = $40.
Pre-Algebra
12-5 Linear Functions
Lesson Quiz
Write the rule for each linear function.
1.
x
y
–2
–1
0
1
2
8
5
2
–1
–4
2.
x
y
–3
0
3
5
7
–10
–1
8
14
20
f(x) = –3x + 2
f(x) = 3x – 1
3. Andre sells toys at the craft fair. He pays $60
to rent the booth. Materials for his toys are $4.50
per toy. Write a function for Andre’s expenses for
the day. Determine his expenses if he sold 25
toys.
f(x) = 4.50x + 60; $172.50
Pre-Algebra