6-1 Multiple Representations of Functions Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2 Holt Algebra 2

Create a table of values for each function. Warm Up Create a table of values for each function. 1. y = 2x – 3 2. f(x) = 4x – x2

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

Essential Question What clues help you determine whether a function is linear, quadratic, or exponential?

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.

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.

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.

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.

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.

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?

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.

Example 2 Continued Step 2 Write an equation. is equal to minus Distance 80 3 meters per second. d = 80 – 3t

Step 3 Find the intercepts and graph the equation. 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 = = 26 –80 –3 2 3 t-intercept: 26 2 3 Janet will reach the shore after seconds. 26 2 3

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.

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.

Enter the data in the STAT table on your calculator.

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. 100 500

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!

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.

Example 3A Continued The data do not appear to be linear, so check finite differences.

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.

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.

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.

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.

The data appears to be exponential, so check for a common ratio. Example 3B Continued 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 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.

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.

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.

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.

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.

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.

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.

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.