University:
High School
Course:
High School Algebra
Academic year:

2022
Views:

459

Pages:

8
Author:

Brent Barnett

2.4 Modeling with Quadratic Functions Question: How can you use a quadratic function to model a real-life situation? EXPLORATION: Modeling with a Quadratic Function Work with a partner. The graph shows a quadratic function of the form P(t) = at² + bt + c which approximates the yearly profits for a company, where P(t) is the profit in year t. a. Is the value of a positive, negative, or zero? Explain. Positive, since it curves up b. Write an expression in terms of a and b that represents the year t when the company made the least profit. ...learning that... c. The company made the same yearly profits in 2004 and 2012. Estimate the year in which the company made the least profit. maybe '08, the middle d. Assume that the model is still valid today. Are the yearly profits currently increasing, decreasing, or constant? Explain. Increasing since it curves up on the right Core Concepts We will cover this Section Writing Quadratic Equations Given a point and the vertex (h, k) Use vertex form: y = a(x - h)² + k Given a point and the x-intercepts, p and q Use intercept form: y = a(x - p)(x - q) Given three points Write and solve a system of three equations in three variables. x + y + z = 4 2x - y + z = 4 -x + 4y - z = 10 Three equations and then solve the system like 2.4 Writing Quadratic Equations Given a point and the vertex (h, k) Use vertex form: y = a(x - h)² + k In Exercises 1-4, write an equation of the parabola in vertex form. 1. Vertex: (2, -2) Point: (4, 6) y = a(x - h)² + k 6 = a(4 - 2)² - 2 6 = a(2)² - 2 6 = 4a - 2 8 = 4a 2 = a Our equation is: y = 2(x - 2)² - 2 2. Vertex: (0, 3) Point: (2, 1) y = a(x - h)² + k 1 = a(2 - 0)² + 3 1 = a(2)² + 3 1 = 4a + 3 -2 = 4a -½ = a So our equation is... y = -½(x)² + 3 or y = -½(x - 0)² + 3 2.4 Writing Quadratic Equations Given a point and the vertex (h, k), use vertex form: y = a(x - h)² + k In Exercises 1-4, write an equation of the parabola in vertex form. 3. passes through (-3, 0) and has vertex (-1, -8) Point: (-3, 0) Vertex: (-1, -8) y = a(x - h)² + k 0 = a(-3 - (-1))² + (-8) 0 = a(-2)² - 8 0 = 4a - 8 8 = 4a 2 = a So the equation is: y = 2(x + 1)² - 8 4. passes through (-4, 7) and has vertex (-2, 5) Point: (-4, 7) Vertex: (-2, 5) y = a(x - h)² + k 7 = a(-4 - (-2))² + 5 7 = a(-2)² + 5 7 = 4a + 5 2 = 4a ½ = a So the equation is: y = ½(x + 2)² + 5