Modeling with Quadratic Functions

The population, y, of a small town can be represented using a quadratic model. The maximum population, 6,000 people, occurred 20 years after record keeping began. Five years later, the population was 5,950. If x represents the number of years after record keeping began, which equation represents the given scenario? y = 2(x - 20)2 + 6,000 y = -(x - 20)2 + 6,000 y = -2(x - 20)2 + 6,000 y = (x - 20)2 + 6,000

C.y = -2(x - 20)2 + 6,000

An engineer sketches a design for a flashlight that uses a mirror in the shape of a parabola to maximize the output of the light. The function representing the mirror is graphed on the left. Which function models the situation? f(x) = (x - 6)2 + 2 f(x) = -(x - 0)2 + 20 f(x) = 3(x - 6)2 + 2 f(x) = -3(x - 0)2 + 20

A. f(x) = (x - 6)2 + 2

The curve produced by the water coming from a hose is sketched onto a graph with zeros at 0 and 5. The point (4, 1) also lies on the curve. If h(x) represents the vertical distance from where the water first comes out of the hose and x represents the horizontal distance, which statements are true? Check all that apply. The scenario can be represented by the function h(x) = -0.25(x)(x - 5). The scenario can be represented by the function h(x) = (x)(x + 5). The vertex is on the line x = 2.5. The greatest height that the water reaches is 1.5 units. The scenario can be represented by the function h(x) = -1(x - 5).

A. The scenario can be represented by the function h(x) = -0.25(x)(x - 5). C. The vertex is on the line x = 2.5.

A soccer ball is kicked into the air, and its path can be modeled with a quadratic function. The table shows the time, t, in seconds, and the height of the soccer ball, h, in feet. Follow the steps to write the function. 1.Identify x-intercepts:(0, 0) and (5, 0)2.Use factored form:f(t) = a(t − 0)(t − 5) f(t) = at(t − 5)3.Find the value of:12 = a(3)(3 − 5)4.Write the function:

f(t)= -2 t(t-5)

An image of a parabolic lens is projected onto a graph. The y-intercept of the graph is (0, 90), and the zeros are 5 and 9. Which equation models the function? y = 90(x - 5)(x - 9) y = 2(x - 5)(x - 9) y = 90(x + 5)(x + 9) y = 2(x + 5)(x + 9)

B. y = 2(x - 5)(x - 9)

A ball was tossed up into the air. The height of the ball as a function of the time the ball is in the air in seconds can be modeled by a quadratic function. At the time the ball was initially thrown in the air, it was 7.5 meters off the ground. After 1.25 seconds, the ball was at its maximum height of 15.3125 meters. What was the height of the ball 2 seconds after being thrown? Round your answer to the nearest tenth as needed.

12.5 meters

Which values complete the function?

f(x)=-1(x-3)(x+4)

The vertex of a quadratic function is located at (1, 4), and the y-intercept of the function is (0, 1). What is the value of a if the function is written in the form y = a(x - h)2 + k? a =

a= -3

Jessica is asked to write a quadratic equation to represent a function that goes through the point (8, -11) and has a vertex at (6, -3). Her work is shown below. -11 = a(8 - 6)2 - 3 -11 = a(2)2 - 3 -11 = 4a - 3 -8 = 4a a = -2 After Jessica gets stuck, she asks Sally to help her finish the problem. Sally states that Jessica needs to write the quadratic equation using the value she found for a, -2, and the point (8, -11). Evaluate Jessica's work and Sally's review to determine which statement is true. Jessica did not solve for the correct value of a, but Sally's review is correct. Jessica did not solve for the correct value of a, and Sally's review is incorrect. Jessica solved for the correct value of a, but Sally's review was incorrect. Both Jessica's work and Sally's review are correct. The equation is f(x) = -2(x - 8)2 - 11.

C. Jessica solved for the correct value of a, but Sally's review was incorrect.