Modeling with Polynomial Functions Section 6.9 Modeling with Polynomial Functions

Using Finite Differences You know that two points determine a line, three points determine a parabola and four points determine the graph of a cubic function. To decide whether y-values for equally-spaced x-values can be modeled by a polynomial function, you can use finite differences.

Finding Finite Differences We find differences the same way we found them in algebra 1. Remember, we are only trying to see if there is a pattern with the sequence of numbers. We must find a constant difference in order to use a polynomial model.

Using Differences If the first difference is constant, the function is a linear model. If the second difference is constant, the function is a quadratic model. If the third difference is constant, the function is a cubic model.

Ex 1a 1.1 Using Differences to Identify Patterns Find the next three terms in the sequence by using constant differences: 1, 3, 5, 7, 9, ... 11 13 15 2 2 2 2 First differences 2 2 2 Add 2 to each term to find the next term: 9 + 2 = 11 11 + 2 = 13 13 + 2 = 15

Ex 1b 1.1 Using Differences to Identify Patterns Find the next three terms in the sequence by using constant differences: 80, 73, 66, 59, 52, ... 45 38 31 -7 -7 -7 -7 First differences -7 -7 -7 Subtract 7 from each term to find the next term: 52 - 7 = 45 45 - 7 = 38 38 - 7 = 31

Ex 2a: Find the next three terms: 1, 4, 9, 16, 25 1.1 Using Differences to Identify Patterns Find the next three terms: 1, 4, 9, 16, 25 Find the first differences. 36 49 64 3 5 7 9 11 13 15 1st differences 2 2 2 2 2 2 2nd differences

Ex 2b: Find the next three terms: 37, 41, 48, 58, 71, … 1.1 Using Differences to Identify Patterns Find the next three terms: 37, 41, 48, 58, 71, … Find the first differences. 87 106 128 4 7 10 13 16 19 22 1st differences 3 3 3 3 3 3 2nd differences

Writing a Cubic Function Write the cubic function whose zeros are -3, 2 and 5, and passes through (0, -15) First, write the function in intercept form: f(x) = a(x + 3)(x – 2)(x – 5) Solve for a by substituting the point in for (x,y) -15 = a(0 +3)(0-2)(0-5) -15 = 30a a = - ½ f(x) = - ½ (x + 3)(x – 2)(x – 5)

Example: Find a polynomial function that gives the nth triangular pyramidal number: F(1) = 1, F(2) = 4, F(3)=10, F(4)=20, F(5)=35, F(6)= 56, F(7) = 84 First, start by finding the constant difference: 1 4 10 20 35 56 84 V V V V V V 3 6 10 15 21 28

1 4 10 20 35 56 84 V V V V V V 3 6 10 15 21 28 V V V V V 3 4 5 6 7 V V V V 1 1 1 1 Since the third differences are constant, we know it can be model by a cubic function.

Cubics are of the form: f(x) = ax3 + bx2 + cx + d Now, substitute the first four terms into the equation to produce a system of four equations in four variables.

Simplify the system: Use matrices on the calculator to solve the system for a, b, c, and d.

Write the equation: a = 1/6, b = ½, c = 1/3, d = 0 So,

You try an example: Write the equation for the following: 1 2 3 4 5 6 F(x) 20 58 122 218 352

You try an example: Write the equation for the following: 1 2 3 4 5 6 F(x) 20 58 122 218 352

Cubic Regressions on Calculator You can also use the graphing calculator to find the equation of a cubic. Put in the data into L1 and L2 (just like linear regressions). Select CubicReg function to find the equation.

Find the polynomial model for the data: X 9 11 13 15 17 19 21.5 Y 6.43 7.61 8.82 9.86 10.88 12.36 15.24

Assignment Section 6.9: page 383 – 385 # 14, 17, 23 – 31 odd, 33, 47