Download 3t Holt McDougal Algebra 2 6-1 Multiple Representations of

6-1
6-1 Multiple
MultipleRepresentations
Representationsof
ofFunctions
Functions
Warm Up
Lesson Presentation
Lesson Quiz
HoltMcDougal
Algebra 2Algebra 2
Holt
6-1 Multiple Representations of Functions
Warm Up
Create a table of values for each
function.
1. y = 2x – 3
2. f(x) = 4x – x2
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Objectives
Translate between the various
representations of functions.
Solve problems by using the various
representations of functions.
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Essential Question
• What clues help you determine whether a
function is linear, quadratic, or
exponential?
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
An amusement park manager
estimates daily profits by
multiplying the number of
tickets sold by 20. This verbal
description is useful, but other
representations of the function
may be more useful.
These different representations can help the manager
set, compare, and predict prices.
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Example 1: Business Application
Sketch a possible graph to represent the
following.
Ticket sales were good
until a massive power
outage happened on
Saturday that was not
repaired until late Sunday.
The graph will show decreased
sales until Sunday.
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Check It Out! Example 1a
What if …? Sketch a possible graph to
represent the following.
The weather was
beautiful on Friday
and Saturday, but it
rained all day on
Sunday and Monday.
The graph will show decreased
sales on Sunday and Monday.
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Check It Out! Example 1b
Sketch a possible graph to represent the
following.
Only of the rides were
running on Friday and
Sunday.
The graph will show decreased
sales on Friday and Sunday.
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Because each representation of a function
(words, equation, table, or graph) describes
the same relationship, you can often use any
representation to generate the others.
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Example 2: Recreation Application
Janet is rowing across an 80-meter-wide river
at a rate of 3 meters per second. Create a table,
an equation, and a graph of the distance that
Janet has remaining before she reaches the
other side. When will Janet reach the shore?
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Example 2 Continued
Step 1 Create a table.
Let t be the time in seconds
and d be Janet’s distance, in
meters, from reaching the
shore.
Janet begins at a distance of
80 meters, and the distance
decreases by 3 meters each
second.
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Example 2 Continued
Step 2 Write an equation.
Distance
d
is equal to 80 minus 3 meters per second.
=
Holt McDougal Algebra 2
80
–
3t
6-1 Multiple Representations of Functions
Example 2 Continued
Step 3 Find the intercepts and graph the equation.
d-intercept:80
Solve for t when d = 0
d = 80 – 3t
0 = 80 – 3t
t = –80 = 26 2
–3
3
2
t-intercept: 26
3
Janet will reach the shore after 26 2 seconds.
3
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Check It Out! Example 2
The table shows the height, in feet, of an
arrow in relation to its horizontal distance
from the archer. Create a graph, an
equation, and a verbal description to
represent the height of the arrow with
relation to its horizontal distance from the
archer.
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Check It Out! Example 2 Continued
Step 1 Use the table to check finite differences.
Height (ft.)
6.55
59.8
90.55 98.80
84.5
47.50
First differences 53.25 30.75 8.25 –14.3 –37
Second differences
–22.5 –22.5 –22.55 –22.7
Since second differences are close to constant, use
the quadratic regression feature to write an equation.
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Enter the data in the STAT table on your calculator.
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Check It Out! Example 2 Continued
Step 2 Write an equation. Use
the quadratic regression feature.
–0.002x2 + 0.86x + 6.52
Step 3 Graph the
equation. The archer
shoots the arrow from a
height of 6.52 ft. It travels
horizontally about 220 ft
to its peak height of about
99 ft, and then it lands on
the ground about 500 ft
away.
100
0
0
Holt McDougal Algebra 2
500
6-1 Multiple Representations of Functions
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Remember!
The point-slope form of the equation of a line is
y – y1 = m(x – x1), where m is the slope and
(x1, y1) is a point on the line. (Lesson 2-4)
Remember!
First differences are constant in linear functions.
Second differences are constant in quadratic
functions. (Lesson 5-9)
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Example 3: Using Multiple Representations to Solve
Problems
A hotel manager knows that the number of
rooms that guests will rent depends on the
price. The hotel’s revenue depends on both
the price and the number of rooms rented.
The table shows the hotel’s average nightly
revenue based on room price. Use a graph
and an equation to find the price that the
manager should charge in order to maximize
his revenue.
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Example 3A Continued
The data do not appear to be linear, so check finite
differences.
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Example 3A Continued
Revenues
21,000
First differences
22,400,
1,400
Second differences
23,400
1,000
400
24,000
600
400
Because the second differences are constant, a
quadratic model is appropriate. Use a graphing
calculator to perform a quadratic regression on
the data.
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Example 3A Continued
The equation y = –2x2 + 440x models the data,
and the graph appears to fit. Use the TRACE or
MAXIMUM feature to identify the maximum
revenue yield. The maximum occurs when the
hotel manager charges $110 per room.
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Example 3B: Using Multiple Representations to Solve
Problems
An investor buys a property for $100,000.
Experts expect the property to increase in value
by about 6% per year. Use a table, a graph and
an equation to predict the number of years it
will take for the property to be worth more than
$150,000.
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Example 3B Continued
Make a table for the property. Because you are
interested in the value of the property, make a graph,
by using years t as the independent variable and
value as the dependent variable.
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Example 3B Continued
The data appears to be exponential, so check for a
common ratio.
r1 =
y1
y0
106,000
=
= 1.06
100,000
r3 =
y3
y2
119,102
=
≈ 1.06
112,360
r2 =
y2
y1
112,360
=
= 1.06
106,000
r4 =
y4
y3
126,248
=
≈ 1.06
119,102
Because there is a common ratio, an exponential
model is appropriate. Use a graphing calculator to
perform an exponential regression on the data.
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Example 3B Continued
The equation y = 100,000(1.06)x models the data and
the graph appears to fit. Use the TRACE feature to
identify the value of x that corresponds to 150,000 for
y. It appears that the property will be worth more than
$150,000 after 7 years.
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Check It Out! Example 3
Bartolo opened a new
sporting goods business and
has recorded his business
sales each week. To break
even, Bartolo needs to sell
$48,000 worth of
merchandise in a week.
Assuming the sales trend
continues, use a graph and
an equation to find the
number of weeks before
Bartolo breaks even.
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Check It Out! Example 3 Continued
The data appears to be exponential, so check for a
common ratio.
r1 =
y1
y0
27,500
=
25,000
r3 =
y3
y2
33,275
=
30,250
= 1.1
= 1.1
r2 =
y2
y1
30,250
=
27,500
= 1.1
r4 =
y4
y3
36,603
=
33,275
≈ 1.1
Because there is a common ratio, an exponential
model is appropriate. Use a graphing calculator to
perform an exponential regression on the data.
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Check It Out! Example 3 Continued
The equation y = 22,727.15(1.1)x models the data and
the graph appears to fit. Use the TRACE feature to
identify the value of x that corresponds to 48,000 for y.
He will break even after 8 weeks.
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Lesson Quiz: Part I
1. The graph shows the number of cars in a high
school parking lot on a Saturday, beginning at
10 A.M. and ending at 8 P.M. Give a possible
interpretation for this graph.
Possible answer: Football
practice goes from 11:00 A.M.
until 1:00 P.M. Families begin
arriving at 4:00 P.M. for a play
that begins at 5:00 P.M. and
ends at 7:00 P.M. After the
play, most people leave.
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Lesson Quiz: Part II
2. An online computer game company has 10,000
subscribers paying $8 per month. Their research
shows that for every 25-cent reduction in their
fee, they will attract another 500 users. Use a
table and an equation to find the fee that the
company should charge to maximize their
revenue.
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
Essential Question
• What clues help you determine whether a
function is linear, quadratic, or
exponential?
• First difference is linear, second difference
is quadratic, and a constant ratio is
exponential.
Holt McDougal Algebra 2
6-1 Multiple Representations of Functions
• Teacher: Would you use a table, graph,
or equation to solve this problem?
• Student: A table …then I could sit down
and eat while I worked.
Holt McDougal Algebra 2