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Holt McDougal Algebra 2

6-1 Multiple Representations of Functions 6-1 Multiple Representations of Functions

Holt Algebra 2

Warm Up

Lesson Presentation

Lesson Quiz

Holt McDougal Algebra 2

Holt McDougal Algebra 2

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

Translate between the various representations of functions. Solve problems by using the various representations of functions.

Objectives

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

Sketch a possible graph to represent the following.

Example 1: Business Application

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.

The graph will show decreased sales on Friday and Sunday.

Only of the rides were running 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 Janets 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. is equal to minus Distance 80 3 meters per second.

d = 80 3t

Holt McDougal Algebra 2

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

0 = 80 3t

t = = 26 80

3 2 3

d = 80 3t

t-intercept: 26 2 3

Janet will reach the shore after seconds. 26 2 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

First differences 53.25 30.75 8.25 14.3 37 Second differences 22.5 22.5 22.55 22.7

Step 1 Use the table to check finite differences.

Height (ft.) 6.55 59.8 90.55 98.80 84.5 47.50

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

Step 2 Write an equation. Use the quadratic regression feature.

Check It Out! Example 2 Continued

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

500 0 0

Holt McDougal Algebra 2

6-1 Multiple Representations of Functions

Holt McDougal Algebra 2

6-1 Multiple Representations of Functions

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)

Remember!

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 hotels revenue depends on both the price and the number of rooms rented. The table shows the hotels 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 22,400, 23,400 24,000

First differences 1,400 1,000 600

Second differences 400 400

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

The data appears to be exponential, so check for a common ratio.

r1 = = = 1.06 y1

y0

106,000 100,000 r2 = = = 1.06

y2

y1

112,360 106,000

r3 = = 1.06 y3

y2

119,102 112,360 r4 = = 1.06

y4

y3

126,248 119,102

Because there is a common ratio, an exponential model is appropriate. Use a graphing calculator to perform an exponential regression on the data.

Example 3B Continued

Holt McDougal Algebra 2

6-1 Multiple Representations of Functions

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.

Example 3B Continued

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 = = = 1.1 y1

y0

27,500 25,000 r2 = = = 1.1

y2

y1

30,250 27,500

r3 = = = 1.1 y3

y2

33,275 30,250 r4 = = 1.1

y4

y3

36,603 33,275

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

The equation y = 22,727.15(1.1)x models th

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