Transcript C3_ch13_04_Seventh Grade Wednesday

13-4
13-4Linear
LinearFunctions
Functions
Warm Up
Problem of the Day
Lesson Presentation
Course
Course
33
13-4 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).
Course 3
2, 11, 6
13-4 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.
Course 3
13-4 Linear Functions
Learn to identify linear functions.
Course 3
13-4 Linear Functions
Vocabulary
linear function
function notation
Course 3
13-4 Linear Functions
A linear function can be described by a linear
equation. You can use function notation to
show that the output value of the function f,
written f(x), corresponds to the input value x.
Reading Math
f(x) is read “f of x.”
f(1)is read “f of 1.”
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.
Course 3
13-4 Linear Functions
Additional Example 1: Identifying Linear Functions
Determine whether the function f(x) = 2x3 is
linear.
f(x) = 2x3
y
4
Graph the function.
2
x
-4
-2
2
-4
Course 3
4
f(x) = 2x3 does not
represent a linear function
because its graph is not in a
straight line.
13-4 Linear Functions
Check It Out: Example 1
Determine whether the function f(x) = -2x + 4
is linear.
f(x) = –2x + 4
y
4
Graph the function.
2
x
-4
-2
2
-2
-4
Course 3
4
f(x) = -2x + 4 does
represent a linear function
because its graph is in a
straight line. It has a slope
of -2 and a y-intercept of 4.
13-4 Linear Functions
Additional Example 2A: Writing the Equation for a
Linear Function
Write a rule for the linear function.
Step 1 Identify the y-intercept
b from the graph.
b=2
Step 2 Locate another point
on the graph, such as (1, 4).
Step 3 Substitute the x- and
y-values of the point into the
equation, f(x) = mx + b, and
solve for m.
Course 3
13-4 Linear Functions
Additional Example 2A Continued
f(x) =
4=
4=
–2
2=
mx + b
m(1) + 2
m+2
–2
m
The rule is f(x) = 2x + 2.
Course 3
(x, y) = (1, 4)
13-4 Linear Functions
Additional Example 2B: Writing the Equation for a
Linear Function
Write a rule for the linear function.
x
y
Step 1 Locate two points.
–3 –8
(1, 4) and (3, 10)
–1 –2
Step 2 Find the slope m.
1
4
3
10
y2 – y1 10 – 4 6
m = x2 – x1 =
= =3
3–1 2
Step 3 Substitute the x- and
y-values of the point into the
equation, f(x) = mx + b, and
solve for b.
Course 3
13-4 Linear Functions
Additional Example 2B Continued
f(x) = mx + b
4 = 3(1) + b
4= 3+b
–3 –3
1=
b
The rule is f(x) = 3x + 1.
Course 3
(x, y) = (1, 4)
13-4 Linear Functions
Check It Out: Example 2A
Write a rule for the linear function.
Step 1 Identify the y-intercept
b from the graph.
y
4
2
-4
-2
2
-4
Course 3
4
b=1
Step 2 Locate another point
x
on the graph, such as (5, 2).
Step 3 Substitute the x- and
y-values of the point into the
equation, f(x) = mx + b, and
solve for m.
13-4 Linear Functions
Check It Out: Example 2A Continued
f(x) = mx + b
2 = m(5) + 1
2 = 5m + 1
–1
–1
1 = 5m
m=1
5
The rule is f(x) = 1 x + 1.
5
Course 3
(x, y) = (5, 2)
13-4 Linear Functions
Check It Out: Example 2B
Write a rule for the linear function.
x
y
Step 1 Locate two points.
0
5
(0, 5) and (1, 6)
1
6
2
7
–1
4
Step 2 Find the slope m.
y2 – y1 6 – 5
m = x2 – x1 = 1 – 0 = 1
1 =1
Step 3 Substitute the x- and
y-values of the point into the
equation, f(x) = mx + b, and
solve for b.
Course 3
13-4 Linear Functions
Check It Out: Example 2B Continued
f(x) = mx + b
5 = 1(0) + b
5= b
(x, y) = (0, 5)
The rule is f(x) = x + 5.
Course 3
13-4 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
Course 3
cost will be f(12) = 1.5(12) + 15 =
18 + 15 = $33.
13-4 Linear Functions
Check It Out: 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 books
2=m
purchases, the cost will be
f(10) = 2(10) + 20 = 20 + 20 = $40.
Course 3
13-4 Linear Functions
Lesson Quiz: Part I
Determine whether each function is linear.
1. f(x) = 4x2
not linear
2. f(x) = 3x + 1
linear
Write the rule for the linear function.
f(x) = 1 x - 1
2
Course 3
13-4 Linear Functions
Lesson Quiz: Part II
Write the rule for each linear function.
2.
x
y
–3
0
3
–10 –1 8
5
7
14 20
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. Find a rule for the linear
function that describes Andre's expenses for
the day. Determine his expenses if he sold 25
toys.
f(x) = 4.50x + 60; $172.50
Course 3