Bellringer Use the FOIL method to solve the following problems. 1. (1 + x)(3 + 2x) 2. (2 + 5x)( x) 3. 2(a 2 + a)(3a 2 + 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) = ax² + bx + c, where a ≠ 0. Quadratic TermLinear TermConstant Term

Determine whether each function is linear or quadratic. Identify the quadratic, linear, and constant terms. a.ƒ(x) = (2x – 1) 2 = (2x – 1)(2x – 1)Multiply. = 4x 2 – 4x + 1Write in standard form. This is a quadratic function. Quadratic term: 4x 2 Linear term: –4x Constant term: 1 Classifying Functions

(continued) b.ƒ(x) = x 2 – (x + 1)(x – 1) = x 2 – (x 2 – 1)Multiply. = 1Write in standard form. This is a linear function. Quadratic term: none Linear term: 0x (or 0) Constant term: 1 Continued

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. Below is the graph of y = x 2 – 6x Identify the vertex and the axis of symmetry. Identify points corresponding to P and Q. Points on a Parabola

Find the quadratic function to model the values in the table. xyxy –2 – –10 Substitute the values of x and y into y = ax 2 + bx + c. The result is a system of three linear equations. y = ax 2 + bx + c The solution is a = –2, b = 7, c = 5. Substitute these values into standard form. The quadratic function is y = –2x 2 + 7x + 5. Using one of the methods of Chapter 3, solve the system 4a – 2b + c = –17 a – b + c = 10 25a + 5b + c = –10 { –17 = a(–2) 2 + b(–2) + c = 4a – 2b + cUse (–2, –17). 10 = a(1) 2 + b(1) + c = a + b + cUse (1, 10). –10 = a(5) 2 + b(5) + c = 25a + 5b + cUse (5, –10). Fitting a Quadratic Function to 3 Points

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 (m) Wave Speed (m/s) Real World Example

(continued) Wavelength (m) Wave Speed (m/s) Step 1:Enter the data. Use QuadReg. Step 2:Graph the data and the function. Step 3:Use the table feature to find ƒ(6). An approximate model of the quadratic function is y = 0.59x x – At a wavelength of 6 meters the wave speed is approximately 23m/s. Continued

Homework 5-1 Pg 241 # 1,2,10,11,16, 19, 21