5-8 Polynomial Models in the Real World

Quick Review

A data set can be modeled by a polynomial function. Methods of finding a model that fits the data include the ( n + 1 ) open n plus 1 close  Point Principle and regression. Linear, quadratic, cubic, and quartic regressions can be performed on a graphing calculator. A higher R 2 r squared  value means a better fit. Once the equation that models the data is known, it can be used to make predictions.

Example

For the data set (8, 30), (10, 45), and (11, 65), predict y when x = 15 . x equals 15 .

Enter 8, 10, and 11 in L1 and 30, 45, and 65 in L2. Choose LINREG to find the regression model y ≈ 11 . 071 x − 60 . 357 . y almost equal to 11 . 071 x minus 60 . 357 .  The r 2 r squared  value is about 0.928.

Now try QUADREG. The model is y ≈ 4 . 17 x 2 − 67 . 5 x + 303 . 3 y almost equal to 4 . 17 , x squared , minus 67 . 5 x plus 303 . 3  with an R 2 r squared  value of 1. Assuming the model makes sense in context, it fits the data better.

Using the quadratic model, when x = 15 , y ≈ 228.3 . x equals 15 comma y almost equal to , 228.3 , .

Exercises

82. Write a polynomial function whose graph passes through (0, 5), (2, 10), and (1, 4). Use a regression to check your answer.

83. Find a linear, a quadratic, and a cubic model for the data. Which model best fits the data?

    x	3	8	15	21
    y	7	11	26	44

84. Use CUBICREG to model the data below. Then use the model to estimate the population in 2008. Let x be the number of years after 2000.

    Year	2004	2007	2009	2010
    Population	457	910	1244	1315

5-9 Transforming Polynomial Functions

Quick Review

A polynomial function can be transformed into other polynomial functions using stretches, reflections, and translations. The monomial function y = a x b y equals , eh x to the b  is called a power function.

Example

This is the graph of a cubic function. Determine which sequence of transformations you can apply to the graph of the parent function y = x 3 y equals , x cubed  to get this graph. Write an equation for the graph.

Translate the parent function 4 units left and 2 units down: y = ( x + 4 ) 3 − 2 y equals . open x plus 4 close cubed . minus 2

Exercises

Determine the cubic function obtained from the parent function y = x 3 y equals , x cubed  after each sequence of transformations.

85. a reflection across the x-axis; a translation 1 unit up; and a translation 2 units right
86. a vertical stretch by a factor of 6; and a translation 3 units left
87. Find a quartic function whose only real zeros are 4 and 6.
88. The parent power function y = x 5 y equals , x to the fifth  is translated 3 units up and is compressed by the factor 0.3. Write the function.

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