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

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

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

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.

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

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 = 0.59x2 + 0.34x – 0.33. At a wavelength of 6 meters the wave speed is approximately 23m/s.