1. Modeling Data With Quadratic Functions
Lesson 5-1
Additional Examples
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.
= 4x2 – 4x + 1
Write in standard form.
This is a quadratic function.
Quadratic term: 4x2
Linear term: –4x
Constant term: 1

2. Modeling Data With Quadratic Functions
Lesson 5-1
Additional Examples
(continued)
b. ƒ(x) = x2 – (x + 1)(x – 1)
= x2 – (x2 – 1)
Multiply.
= 1
Write in standard form.
This is a linear function.
Quadratic term: none
Linear term: 0x (or 0)
Constant term: 1

3. Modeling Data With Quadratic Functions
Lesson 5-1
Additional Examples
Below is the graph of y = x2 – 6x Identify the vertex and the axis of symmetry. Identify points corresponding to P and Q.
The vertex is (3, 2).
Corresponding point Q (2, 3) is one unit to the left of the axis of symmetry.
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.

4. Modeling Data With Quadratic Functions
Lesson 5-1
Additional Examples
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.

5. Modeling Data With Quadratic Functions
Lesson 5-1
Additional Examples
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) 3 5 7 8
Wave Speed
(m/s) 6 16 31 40

6. Modeling Data With Quadratic Functions
Lesson 5-1
Additional Examples
(continued)
Wavelength
(m) 3 5 7 8
Wave Speed
(m/s) 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 ƒ(6).
An approximate model of the quadratic function is y = 0.59x x – 0.33.
At a wavelength of 6 meters the wave speed is approximately 23m/s.