Transcript Chapter 4 Functions Notes

Representing
Relationships
Chapter 4, Lesson 1
Real-World Link
To achieve orbit, the space shuttle
must travel at a rate of about 5
miles per second. The table shows
the total distance d that the craft
covers in certain periods of time t.
a. Write an algebraic expression for the distance
in miles for any number of seconds t.
5t
b. Describe the relationship in words.
The distance is 5 times the number of seconds.
Real-World Link
To achieve orbit, the space shuttle
must travel at a rate of about 5
miles per second. The table shows
the total distance d that the craft
covers in certain periods of time t.
c. Graph the ordered pairs.
Describe the shape of the
graph.
Linear Equation
An equation with a graph that makes a
straight line.
Can have more than one variable.
Example 1
Write an equation to find the number of liters in any
number of quarts. Describe the relationship in
words.
The rate of change is 0.95.
EQUATION:
ℓ = 0.95q
IN WORDS:
The rate of change between quarts and liters is
0.95.
Example 2
About how many liters are in 8 quarts?
EQUATION:
ℓ = 0.95q
ℓ = 0.95(8)
ℓ = 7.6
There are about 7.6 liters in 8 quarts.
Got it? 1 & 2
The total cost of tickets to the school
play is shown in the table.
a. Write an equation to find the
total cost of any number of tickets.
Describe the relationship in words.
b. Use the equation to find the cost of 15 tickets.
Example 3
The total distance Marlon ran one week is shown
by the graph.
Write an equation to find the
number of miles run y after
any number of days x.
1. Find the rate of change.
m=
2.
𝟏𝟒 −𝟕
𝟒 −𝟐
𝟕
𝟐
= = 𝟑. 𝟓
Find the y-intercept.
y = mx + b
y = 3.5x + b
7 = 3.5(2) + b
0=b
The slope is 3.5 and
the y-intercept is 0.
The equation is
y = 3.5x.
Example 4
Using the same equation in Example3, how many
miles will Marion run after 2 weeks?
y = 3.5x
Let x by 14, since x is in days.
y = 3.5(14)
y = 49
Marion will run 49 miles in 2 weeks.
Got it?
The number of trees saved by recycling paper is shown.
a. Write an equation to
find the total number
of trees y that can be
saved for any number
of tons of paper x.
b. Use the equation to
find how many trees
could be saved if 500
tons of paper are
recycled.
Multiple Representations of Linear Equations
Words: The number of trees is equal to 17 times
the number of tons of paper.
Equation: y = 17x
Table:
x
1
2
y
17
34
3
4
51
68
Graph:
Example 5
Chloe completes in jump rope competitions. Her
average rate is 225 jumps per minute.
a. Write an equation to find the number of jumps
in any of amount of minutes.
j = 225m
b. Make a table with 1, 2, 3, 4, and 5 minutes.
Graph the points.
m
225m
j
1
225(1)
225
2
225(2)
450
3
225(3)
675
4
225(4)
900
5
225(5) 1,125
Got it?
Paul earns $7.50 an hour working at a grocery
store.
a. Write an equation the find the amount of
money Paul earned m for any number of hours
h.
b. Make a table to find the earnings if he works 5,
6, 7, and 8 hours. Graph the coordinate points.
Be sure to complete
the Guided Practice
and Journal.
Relations
Chapter 4, Lesson 2
Relations: a set of ordered pairs
Example 1
Express the relation {(2, 6), (-4, 8), (-3, 6), (0, -4)} as
a table and a graph. State the domain and
range.
Domain:
{-4, -3, 0, 2}
Range:
{-8, -4, 6)
Got it?
Express the relation {(-5, 2), (3, -1), (6, 2), (1, 7)} as a
table and a graph. State the domain and range.
Example 2
It cost $3 per hour to
park at the Wild Wood
Amusement Park.
Make a table using x
and y coordinates that
represent the total
cost for 3, 4, 5, and 6
hours.
Graph the ordered
pairs.
Got it? 2
A movie rental store
charges $3.95 per
movie rental. Make a
table using x and y
coordinates for the
total cost of 1,2, 3, and
4 movies.
Graph the ordered
pairs.
Be sure to complete
the Guided Practice
and Journal.
Functions
Chapter 4, Lesson 3
Vocabulary
Function:
Where every domain (input) is matched up with
exactly one range (output)
Example:
m = 14p
If m represent the amount of money you earn,
and p is the number of pizza’s you deliver.
How much money will you make? Depends on
how many pizza’s you deliver.
m = dependent
p = independent
Independent
Variable
Equation
Dependent
Variable
number of
downloads
The equation c = 0.99n
represents the total cost
c for n music
downloads.
cost
number of
hours
The equation d = 4.5h
represents the number
of miles Amber can run
in h hours.
number of
mile
number of
goals
The equation s = g + 3
represents the final
score of games s after g
goals in the final period.
final score
Functions
Example 1:
Find f(-3) if f(x) = 2x + 1.
f(x) = 2x + 1
f(-3) = 2(-3) + 1
f(-3) = -6 + 1
So, f(-3) = -5.
Function Tables:
A way to organize the domain, range and rule on
a table.
Independent Variable = Domain
Dependent Variable = Range
Example 2:
Choose four values for x
and make a function
table for f(x) = x + 5. Then
state the domain and
range.
Domain is {-2, -1, 0, 1}
Range is {3, 4, 5, 6}
Got it? 1 & 2
Choose four values for x to complete the function
table for the function f(x) = x – 7. Then state the
domain and range.
Example 3
There are approximately 770 peanuts in a jar of
peanut butter. The total number of peanuts p(j) is
a function of the number of jars of peanut butter j.
Identify the independent and dependent
variable.
Ask: How many peanuts p(j) are there?
It depends…
So, p(j) or the number of peanuts are the
dependent variable.
Logic tells us that the number of jars is the
independent variable.
Example 4
There are approximately 770 peanuts in a jar of
peanut butter. The total number of peanuts p(j) is
a function of the number of jars of peanut butter j.
What values of the domain and range make
sense for this situation?
DOMAIN:
Ask: positive or negative numbers, whole,
decimals?
only positive whole numbers
RANGE:
The range depends on the x-values, and since
there are 770 peanuts in each jar, the range will
be multiples of 770.
Example 4
There are approximately 770 peanuts in a jar of
peanut butter. The total number of peanuts p(j) is
a function of the number of jars of peanut butter j.
Write a function to represent the total number of
peanuts.
p(j) = 770j
Example 5
There are approximately 770 peanuts in a jar of
peanut butter. The total number of peanuts p(j) is
a function of the number of jars of peanut butter j.
How many peanuts are there in 7 jars of peanut
butter?
p(j) = 770j
p(7) = 770(7)
p(7) = 5,390
There will be 5,390 peanuts in 7 jars of peanut
butter.
The domain is the set of
numbers for the
independent variable.
The range is the set of
numbers for the
dependent variable.
Got it? 3-5
A scrapbooking store is selling rubber stamps for
$4.95 each. The total sales f(n) is a function of the
number of rubber stamps n sold.
a. Identify the independent and dependent
variable.
b.
What values of the domain and range make
sense for this situation?
c.
Write a function equation to represent total
sales.
Determine the total cost of 5 stamps.
d.
Be sure to complete
the Guided Practice
and Journal.
Linear
Functions
Chapter 4, Lesson 4
Sometimes functions are written
with two variables, x and y.
x represents the domain
y represents the range
Example 1
The school stores buys book covers for $2 each and
notebooks for $1. Toni has $5 to spend. The
function y = 5 – 2x represents this situation. Graph
the function and interpret the points graphed.
Chose values for x and
substitute them to find y.
Graph the ordered pairs.
0
1
2
3
Toni has 4 options at the book store.
5 notebooks, 1 cover and 3 notebooks,
or 2 covers and 1 notebook.
Got it? 1
The farmer’s market sells apples for $2 per pound
and pears for $1 per pound. Mallory has $10 to
spend. The function y = 10 – 2x represents this
situation. Graph this function and interpret the
points.
Example 2
Graph y = x + 2.
Make a function table.
Graph the ordered pairs.
Got it? 2
Graph these functions.
a.
y=x–5
b.
y = -2x
Representing Functions
Equation: y = x – 1
Words: the value of y
is one less than the
corresponding value
of x.
Table:
Graph:
Linear Functions: a function where the graph is a
line.
Example: y = mx + b
Continuous vs. discrete data
Continuous – no space between data values
Discrete – have space between data values
Example 3
Each person that enters the store receives a coupon
for $5 off his or her entire purchase.
Write a function to y = 5x
represent the total
value of coupons
given out.
b. Make a function table
for 5, 10, 15, and 20
and graph the points.
c. Is the function
continuous or
discrete? Explain.
There can only be a whole
number of customers, so the
graph is discrete.
a.
x
5x
y
5
5(5)
25
10
5(10)
50
15
5(15)
75
20
5(20)
100
Got it? 3
A store sells trail mix for $5.95 per pound.
a.
b.
c.
Write a function to
represent the total cost
of any number of
pounds.
Make a function table
for 1, 2, 3, 4, and 5
pounds and graph the
points.
Is the function
continuous or discrete?
Explain.
x
1
2
3
4
5
5.95x
y
Make ordered pairs
from the x-value and
y-value. Then graph
the coordinates and
draw a line through, IF
the function is
continuous.
Be sure to complete
the Guided Practice
and Journal.
Compare
Properties of
Functions
Chapter 4, Lesson 5
Real-World Link
Carlos and Stephanie are members to the science
museum. Carlos’s members can be represented by the
function c = 9.99. The cost of Stephanie’s membership
is shown by the table.
Months Cost ($)
a. Make a table to
represent the cost of
Carlos’s membership.
b. Describe the rate of
change for each
function.
Carlos has a rate of 0, and
Stephanie has a rate of 5.
1
5
2
10
3
15
4
20
Months Cost ($)
1
9.99
2
9.99
3
9.99
4
9.99
Example 1
A zebra’s main predator is a lion. Lions can run at a
speed of 53 feet per second over short distances. The
graph shows the speed of a zebra. Compare their
speeds.
Lion’s rate of change = 53
Find the zebra’s rate of change
𝟏𝟏𝟖 − 𝟓𝟗
= 𝟓𝟗
𝟐 −𝟏
The zebra has a faster
rate than the lion.
Got it? 1
A 2013 Ford Acura has a gas mileage of 21 miles per
gallon. The gas mileage of a 2013 Audi is represented
by this graph. Compare their gas mileage.
Example 2
The function m = 140h, where m is the miles traveled in h
hours, represents the speed of the first Japanese high
speed train. The speed today’s high speed train in China is
shown by the table.
a. Compare the functions’ y-intercepts
and rate of change.
They both have the same y-intercept: 0
The rate of change for Japan is 140.
The rate of change for
China is 217.
China’s high speed train is
faster than Japan’s train.
Example 2
The function m = 140h, where m is the miles traveled in h
hours, represents the speed of the first Japanese high
speed train. The speed today’s high speed train in China is
shown by the table.
b. If you ride each train for 5
hours, how far will you
travel on each?
Japan: y = 140h
y = 140(5)
y = 700
You will travel 700 miles on
Japan’s train.
You will travel 1,085
miles in 5 hours on the
Chinese train.
Got it? 2
The number of new movies Movie Madness receives can
be represented by the function m = 7w + 2, where m
represents the number of movies and w represents the
number of weeks. The number of games Game Gallery
receives is shown in the table.
a. Compare the functions’
y-intercept and rate of change.
b. How many new movies will each store have in Week 6?
Example 3
Angela and Ben each have a monthly cell phone bill.
Angela’s monthly bill is represented by y = 0.15x + 49,
where x represents the amount of minutes and y
represents the cost. Ben’s monthly cost is shown by the
graph.
a. Compare the y-intercepts and rate of change.
Angela’s y-intercept is 49 and
the rate of change is 0.15.
Ben’s y-intercept is 60.
𝟖𝟎 −𝟔𝟎
Rate of change =
= 𝟎. 𝟏𝟎
𝟐𝟎𝟎−𝟎
So, Angela pays more per
minute.
Example 3
Angela and Ben each have a monthly cell phone bill.
Angela’s monthly bill is represented by y = 0.15x + 49,
where x represents the amount of minutes and y
represents the cost. Ben’s monthly cost is shown by the
graph.
b. What will be the monthly cost for Angela and
Ben for 200 minutes?
Angela
y = 0.15(200) + 49
y = 79
Angela will pay $79 for 200
minutes.
According to the graph, Ben
will pay $80 for 200 minutes.
Got it? 3
Mandy and Sarah each have a membership to the
gym. Mandy’s membership is represented by the
function y = 3x + 29, where x represents the hours with a
trainer and y represents the cost. The cost of Sarah’s
membership is shown in the graph.
a. Compare the y-intercepts and rates of change.
b. What will be the total cost for Mandy and Sarah
if they each have 4 hours with a trainer?
Example 4
Lorena’s mother needs to rent a truck to move some
furniture. The cost to rent a truck from two different
companies are shown with a graph and table. Which
company should she use if she wants to rent the truck
for 40 miles?
Ron’s Rentals
will charge
$100 for 40
miles.
Cross Town Movers:
slope = 2 and y-intercept is 30
y = 0.5x + 30
It will cost $70 for 40 miles.
Be sure to complete
the Guided Practice
and Journal.
Construct
Functions
Chapter 4, Lesson 6
Real-life Link
Dylan is planning a birthday party at a skating rink.
The rink charges a party fee plus an additional
charge for each guest.
a. Choose two points and find the
rate of change.
(3, 59) and (5, 65)
The rate of change is 3.
b. Write a function to this situation.
y = 3x + 50
c. Graph the ordered pairs and
extend the line until you reach the
y-axis. How much is the party fee?
The party fee is $50.
Example 1
A shoe store offers free points when you sign up for
their rewards card. Then, for each pair of shoes
purchased, you earn an additional number of
points. The graph shows the total points earned
for several pairs of shoes. Find and interpret the
rate of change.
𝐜𝐡𝐚𝐧𝐠𝐞 𝐢𝐧 𝐩𝐨𝐢𝐧𝐭𝐬
𝟗𝟎 − 𝟔𝟎
=
𝐜𝐡𝐚𝐧𝐠𝐞 𝐢𝐧 𝐩𝐚𝐢𝐫𝐬
𝟒−𝟐
= 𝟏𝟓 𝐩𝐨𝐢𝐧𝐭𝐬 𝐩𝐞𝐫 𝐩𝐚𝐢𝐫
You will earn 15 points for
every pair of shoes you
buy.
Got it? 1
Meyer Music charges a yearly subscription fee plus
a monthly fee. The total cost for different number
of months, including the yearly fee, is shown by the
graph. Find and interpret the rate of change and
the initial value.
Example 2
Joan has some photos in her photo album. Each
week she plans to add 12 photos. Joan had 120
photos after 8 weeks. Assume the relationship is
linear. Find and interpret the rate of change and
initial value.
The phrase “each week she adds 12” means the
rate of change is 12.
One of the points on the line is (8, 120)
y = mx + b
120 = 12(8) + b
120 = 96 + b
24 = b
The y-intercept is 24.
So, the initial amount of photos is 24.
Got it? 2
A zoo charges a rental fee plus $2 per hour for
strollers. The total cost of 5 hours is $13. Assume
the relationship is linear. Find the interpret the rate
of change and initial value.
Example 3
The table shows how much money Ava has saved.
Assume the relationship is linear. Find and interpret
the rate of change and initial value.
1. Choose two points to find the
rate of change.
𝟏𝟓𝟎 − 𝟏𝟏𝟎
= 𝟐𝟎
𝟓−𝟑
Ava saves $20 each month.
2. Use slope-intercept form to
find the initial value.
y = mx + b
110 = 20(3) + b
50 = b
Ava initially saved $50.
Got it?
The table shows the monthly cost of sending text
messages. Assume the relationship is linear. Find
and interpret the rate of change and initial value.
# of Messages
(x)
$ Cost (y)
5
10.50
6
10.60
7
10.70
Be sure to complete
the Guided Practice
and Journal.
Linear and
Nonlinear
Functions
Chapter 4, Lesson 7
Real-World Link
The table shows the approximate height and
horizontal distance traveled by a football kicked
at an angle of 30 with an initial velocity of 30 yards
per second.
a. Did the football travel the same
height each half-second?
No
b. Did the football travel the same
length each half-second?
Yes
Real-World Link
The table shows the approximate height and
horizontal distance traveled by a football kicked
at an angle of 30 with an initial velocity of 30 yards
per second.
c. Graph the ordered pairs (time,
height) and (time, length).
Example 1
Determine if each table represents a linear or
nonlinear function. Explain.
a.
As x increases by 2, y decreases by 15 each
time. The table is predictable, so the function is
linear.
Example 1
Determine if each table represents a linear or
nonlinear function. Explain.
b.
As x increases by 3, y increases by different
amounts each time. The rate of change is not
constant. The function in nonlinear.
Got it? 1
Determine whether each table represents a linear
or nonlinear function. Explain.
a.
b.
Example 2
Use the table to determine whether the minimum
number of Calories a tiger cub should eat is linear
function of its age in weeks.
Find the rates.
1000 – 825 = 175
1185 – 1000 = 185
1320 – 1185 = 135
1420 – 1320 = 100
The rates of change are not consistent. The
function in nonlinear.
Got it? 2
Tickets at a school dance cost $5 per student. Are
the ticket sales a linear or nonlinear function of the
number of tickets sold. Explain.
Example 3
A square has a side length of s inches. The area of
the square is a function of the side length. Does
this situation represents a linear or nonlinear
function. Explain.
Make a table to show the area of the square for
side lengths of 1, 2, 3, 4, and 5 inches.
Example 3
A square has a side length of s inches. The area of
the square is a function of the side length. Does
this situation represents a linear or nonlinear
function. Explain.
Graph the function.
Got it? 3
A square has a side length of s inches. The
perimeter of the square is a function of the side
length. Does this situation represent a linear or
nonlinear function. Explain.
Be sure to complete
the Guided Practice
and Journal.
Quadratic
Functions
Chapter 4, Lesson 8
A special type of nonlinear function is a quadratic
function.
The greatest power on the variable is 2.
Examples:
y = x2 + 3x – 5
y = 2x2 + 2x + 1
The graph is U-shaped, opening upward or
downward.
A function in which the
greatest power of the
variable is 2.
The graph is a
straight line.
Quadratic
Functions
The graph always opens
downward.
The graph of a
quadratic function
sometimes open
upward.
Example 1
Graph y = x2.
Example 2
Graph y = -x2 + 4.
Got it? 1&2
Graph y = 6x2.
Example 3
The function d = 4t2 represents the distance d in feet
that a race car will travel over t seconds with a
constant acceleration of 8 feet per second. Graph the
function. Use the graph to find how much time it will
take for the race car to travel 200 feet.
Example 4
The function h = 0.66d2 represents the distance d in miles
you can see from height of h feet. Graph this function.
Then use the graph to estimate how far you can see from a
hot air balloon 1,000 feet in the air.
Distance cannot be negative, so use only positive numbers.
Be sure to complete
the Guided Practice
and Journal.
Qualitative
Graphs
Chapter 4, Lesson 9
Real-World Link
Emily is downloading photos from her digital camera to her
computer. The table shows the percent of photos
downloaded for several seconds.
a. During which period(s) of
time did the percent
downloaded not change?
between 4 and 6 seconds
and between 8 and 10
seconds
b. During which period of
time did the percent
downloaded change the
most?
between 6 and 8 seconds
Real-World Link
Emily is downloading photos from her digital camera to her
computer. The table shows the percent of photos
downloaded for several seconds.
c. Graph and connect the
ordered pairs.
Qualitative Graphs:
Graphs used to represent situations that may not
have numbers.
Example 1
The graph displays the water level in a kiddy pool
that has a drain. Describe the change in the water
level over time.
The water level increased at a constant rate, then
the water was turned off. After some time, the pool
is drained at a constant rate until the water is gone.
Got it? 1
The graph displays the revenue from a local
clothing store. Describe the sales over time.
Example 2
A tennis ball is dropped onto the floor. On each
successive bounce, it rebounds to a height less than its
previous bounce height until it comes to rest on the
floor. Sketch a qualitative graph.
Draw the axis and label.
Sketch the situation.
Example 3
You swing on a swing. Sketch a qualitative graph to
represent the situation.
Draw and label the axes.
Sketch the shape.
Got it? 3&4
A car is traveling at a constant speed. The car
slows down steadily to come to rest at a stop light.
Sketch a qualitative graph to represent the
situation.
Be sure to complete
the Guided Practice
and Journal.