# 5-1 Modeling Data with Quadratic Functions Bellringer Use the FOIL method to solve the following problems. 1. (1 + x)(3 + 2x) 2. (2 + 5x)(11 + 12x) 3. 2(a2 + a)(3a2 + 6a) 5-1 Modeling Data with Quadratic Functions Objectives Quadratic Functions and Their Graphs

Using Quadratic Models Vocabulary A quadratic function is a function that can be written in the standard form f(x) = ax + bx + c, where a 0. Quadratic Term Linear Term Constant Term Classifying Functions Determine whether each function is linear or quadratic. Identify the quadratic, linear, and constant terms.

a. (x) = (2x 1)2 = (2x 1)(2x 1) = 4x2 4x + 1 Multiply. Write in standard form. This is a quadratic function. Quadratic term: 4x2 Linear term: 4x Constant term: 1 Continued (continued)

b. (x) = x2 (x + 1)(x 1) = x2 (x2 1) =1 This is a linear function. Quadratic term: none Linear term: 0x (or 0) Constant term: 1 Multiply. Write in standard form. Points on a Parabola Below is the graph of y = x2 6x + 11. Identify the vertex and the axis of symmetry. Identify points corresponding to P and Q. The vertex is (3, 2).

The axis of symmetry is x = 3. P(1, 6) is two units to the left of the axis of symmetry. Corresponding point P (5, 6) is two units to the right of the axis of symmetry. Q(4, 3) is one unit to the right of the axis of symmetry. Corresponding point Q (2, 3) is one unit to the left of the axis of symmetry. Fitting a Quadratic Function to 3 Points Find the quadratic function to model the values in the table. x y

2 17 1 10 5 10 Substitute the values of x and y into y = ax2 + bx + c. The result is a system of three linear equations. y = ax2 + bx + c 17 = a(2)2 + b(2) + c = 4a 2b + c Use (2, 17).

10 = a(1)2 + b(1) + c = a + b + c Use (1, 10). 10 = a(5)2 + b(5) + c = 25a + 5b + c Use (5, 10). Using one of the methods of Chapter 3, solve the system 4a 2b + c = 17 a b + c = 10 25a + 5b + c = 10 The solution is a = 2, b = 7, c = 5. Substitute these values into standard form. The quadratic function is y = 2x2 + 7x + 5. { Real World Example The table shows data about the wavelength x (in meters) and

the wave speed y (in meters per second) of deep water ocean waves. Use the graphing calculator to model the data with a quadratic function. Graph the data and the function. Use the model to estimate the wave speed of a deep water wave that has a wavelength of 6 meters. Wavelength Wave Speed (m) (m/s) 3 5 7 8 6 16

31 40 Continued (continued) Wavelength (m) Wave Speed (m/s) 3 5 7

8 6 16 31 40 Step 1: Enter the data. Use QuadReg. Step 2: Graph the data and the function. Step 3: Use the table feature to find (6).

An approximate model of the quadratic function is y = 0.59x2 + 0.34x 0.33. At a wavelength of 6 meters the wave speed is approximately 23m/s. Homework 5-1 Pg 241 # 1,2,10,11,16, 19, 21