# holt algebra 2 6-9 curve fitting with polynomial functions 6-9 curving fitting with polynomial...

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• Slide 1
• Holt Algebra 2 6-9 Curve Fitting with Polynomial Functions 6-9 Curving Fitting with Polynomial Functions Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz
• Slide 2
• Holt Algebra 2 6-9 Curve Fitting with Polynomial Functions Warm Up 1. For f(x) = x 3 + 5, write the rule for g(x) = f(x 1) 2 and sketch its graph. g(x) = (x 1) 3 + 3 2. Write a function that reflects f(x) = 2x 3 + 1 across the x-axis and shifts it 3 units down. h(x) = 2x 3 4
• Slide 3
• Holt Algebra 2 6-9 Curve Fitting with Polynomial Functions Use finite differences to determine the degree of a polynomial that will fit a given set of data. Use technology to find polynomial models for a given set of data. Objectives
• Slide 4
• Holt Algebra 2 6-9 Curve Fitting with Polynomial Functions
• Slide 5
• Holt Algebra 2 6-9 Curve Fitting with Polynomial Functions Use finite differences to determine the degree of the polynomial that best describes the data. Example 1A: Using Finite Differences to Determine Degree The x-values increase by a constant 2. Find the differences of the y-values. x468101214 y 2 4.38.310.511.411.5 y 2 4.38.310.511.411.5 First differences: 6.3 4 2.2 0.9 0.1 Not constant Second differences: 2.3 1.8 1.3 0.8 Not constant The third differences are constant. A cubic polynomial best describes the data. Third differences: 0.5 0.5 0.5 Constant
• Slide 6
• Holt Algebra 2 6-9 Curve Fitting with Polynomial Functions Use finite differences to determine the degree of the polynomial that best describes the data. Example 1B: Using Finite Differences to Determine Degree The x-values increase by a constant 3. Find the differences of the y-values. x630369 y916264178151 y916264178151 First differences: 25 10 15 37 73 Not constant Second differences: 15 5 22 36 Not constant The fourth differences are constant. A quartic polynomial best describes the data. Third differences: 20 17 14 Not constant Fourth differences: 3 3 Constant
• Slide 7
• Holt Algebra 2 6-9 Curve Fitting with Polynomial Functions Check It Out! Example 1 Use finite differences to determine the degree of the polynomial that best describes the data. The x-values increase by a constant 3. Find the differences of the y-values. x121518212427 y 3 2329 3143 y 3 2329 3143 Second differences: 14 6 2 10 Not constant The third differences are constant. A cubic polynomial best describes the data. Third differences: 8 8 8 Constant First differences: 20 6 0 2 12 Not constant
• Slide 8
• Holt Algebra 2 6-9 Curve Fitting with Polynomial Functions Example 2: Using Finite Differences to Write a Function The table below shows the population of a city from 1960 to 2000. Write a polynomial function for the data. Step 1 Find the finite differences of the y-values. Year 19601970198019902000 Population (thousands) 4,2675,1856,1667,83010,812 Second differences: 63 683 1318 Third differences: 620 635 Close First differences: 918 981 1664 2982
• Slide 9
• Holt Algebra 2 6-9 Curve Fitting with Polynomial Functions Example 2 Continued Step 2 Determine the degree of the polynomial. Because the third differences are relatively close, a cubic function should be a good model. Step 3 Use the cubic regression feature on your calculator. f(x) 104.58x 3 283.85x 2 + 1098.34x + 4266.79
• Slide 10
• Holt Algebra 2 6-9 Curve Fitting with Polynomial Functions Check It Out! Example 2 The table below shows the gas consumption of a compact car driven a constant distance at various speed. Write a polynomial function for the data. Step 1 Find the finite differences of the y-values. Speed2530354045505560 Gas (gal)23.82525.22525.42730.637 Second differences: 1 0.4 0.6 1.2 2 2.8 Third differences: 0.6 1 0.6 0.8 0.8Close First differences: 1.2 0.2 0.2 0.4 1.6 3.6 6.4
• Slide 11
• Holt Algebra 2 6-9 Curve Fitting with Polynomial Functions Check It Out! Example 2 Continued Step 2 Determine the degree of the polynomial. Because the third differences are relatively close, a cubic function should be a good model. Step 3 Use the cubic regression feature on your calculator. f(x) 0.001x 3 0.113x 2 + 4.134x - 24.867
• Slide 12
• Holt Algebra 2 6-9 Curve Fitting with Polynomial Functions Often, real-world data can be too irregular for you to use finite differences or find a polynomial function that fits perfectly. In these situations, you can use the regression feature of your graphing calculator. Remember that the closer the R 2 -value is to 1, the better the function fits the data.
• Slide 13
• Holt Algebra 2 6-9 Curve Fitting with Polynomial Functions Example 3: Curve Fitting Polynomial Models The table below shows the opening value of a stock index on the first day of trading in various years. Use a polynomial model to estimate the value on the first day of trading in 2000. Step 1 Choose the degree of the polynomial model. Let x represent the number of years since 1994. Make a scatter plot of the data. The function appears to be cubic or quartic. Use the regression feature to check the R 2 - values. Year199419951996199719981999 Price (\$)6836529481306863901 cubic: R 2 0.5833 quartic: R 2 0.8921 The quartic function is more appropriate choice.
• Slide 14
• Holt Algebra 2 6-9 Curve Fitting with Polynomial Functions Step 2 Write the polynomial model. The data can be modeled by f(x) = 32.23x 4 339.13x 3 + 1069.59x 2 858.99x + 693.88 Step 3 Find the value of the model corresponding to 2000. 2000 is 6 years after 1994. Substitute 6 for x in the quartic model. Example 3 Continued f(6) = 32.23(6) 4 339.13(6) 3 + 1069.59(6) 2 858.99(6) + 693.88 Based on the model, the opening value was about \$2563.18 in 2000.
• Slide 15
• Holt Algebra 2 6-9 Curve Fitting with Polynomial Functions Check It Out! Example 3 The table below shows the opening value of a stock index on the first day of trading in various years. Use a polynomial model to estimate the value on the first day of trading in 1999. Step 1 Choose the degree of the polynomial model. Let x represent the number of years since 1994. Make a scatter plot of the data. The function appears to be cubic or quartic. Use the regression feature to check the R 2 - values. Year199419951996200020032004 Price (\$)37543835511711,497834210,454 cubic: R 2 0.8624 quartic: R 2 0.9959 The quartic function is more appropriate choice.
• Slide 16
• Holt Algebra 2 6-9 Curve Fitting with Polynomial Functions Check It Out! Example 3 Continued Step 2 Write the polynomial model. The data can be modeled by f(x) = 19.09x 4 377.90x 3 + 2153.24x 2 2183.29x + 3871.46 Step 3 Find the value of the model corresponding to 1999. 1999 is 5 years after 1994. Substitute 5 for x in the quartic model. f(5) = 19.09(5) 4 377.90(5) 3 + 2153.24(5) 2 2183.29(5) + 3871.46 Based on the model, the opening value was about \$11,479.76 in 1999.
• Slide 17
• Holt Algebra 2 6-9 Curve Fitting with Polynomial Functions Lesson Quiz: Part I 1. Use finite differences to determine the degree of the polynomial that best describes the data. cubic x81012141618 y7.21.28.319.12935.8
• Slide 18
• Holt Algebra 2 6-9 Curve Fitting with Polynomial Functions 2. Lesson Quiz: Part II f(x) = 7.08x 4 126.92x 3 + 595.95x 2 241.81x + 2780.54; about \$3003.50 The table shows the opening value of a stock index on the first day of trading in various years. Write a polynomial model for the data and use the model to estimate the value on the first day of trading in 2002. Year199419961998200020012004 Price (\$)281436035429396241173840