Lesson 24

Using Quadratic Equations to Model Situations and Solve Problems

  • Let’s analyze a situation modeled by a quadratic equation.

Problem 1

The function \(h\) represents the height of an object \(t\) seconds after it is launched into the air. The function is defined by \(h(t)=\text-5t^2+20t+18\). Height is measured in meters.

Answer each question without graphing. Explain or show your reasoning.

  1. After how many seconds does the object reach a height of 33 meters?
  2. When does the object reach its maximum height?
  3. What is the maximum height the object reaches? 

Problem 2

The graphs that represent a linear function and a quadratic function are shown here.

Graph of a linear function and a quadratic function.

The quadratic function is defined by \(2x^2 - 5x\).

Find the coordinates of \(R\) without using graphing technology. Show your reasoning.

Problem 3

Diego finds his neighbor's baseball in his yard, about 10 feet away from a five-foot fence. He wants to return the ball to his neighbors, so he tosses the baseball in the direction of the fence.

Function \(h\), defined by \(h(x)=\text-0.078x^2+0.7x+5.5\), gives the height of the ball as a function of the horizontal distance away from Diego.

Does the ball clear the fence? Explain or show your reasoning.

Problem 4

Clare says, “I know that \(\sqrt3\) is an irrational number, so its decimal never repeats or terminates. I also know that \(\frac29\) is a rational number, so its decimal repeats or terminates. But I don’t know how to add or multiply these decimals, so I am not sure if \(\sqrt3 + \frac29\) and \(\sqrt3 \boldcdot \frac29\) are rational or irrational."

  1. Here is an argument that explains why \(\sqrt3 + \frac29\) is irrational. Complete the missing parts of the argument.

    1. Let \(x = \sqrt3 + \frac29\). If \(x\) were rational, then \(x - \frac29\) would also be rational because . . . .
    2. But \(x - \frac29\) is not rational because . . . .
    3. Since \(x\) is not rational, it must be . . . .
  2. Use the same type of argument to explain why \(\sqrt3 \boldcdot \frac29\) is irrational.
(From Unit 7, Lesson 21.)

Problem 5

The following expressions all define the same quadratic function.

\(x^2+2x-8\)

\((x+4)(x-2)\)

\((x+1)^2-9\)

  1. What is the \(y\)-intercept of the graph of the function?
  2. What are the \(x\)-intercepts of the graph?
  3. What is the vertex of the graph?
  4. Sketch a graph of the quadratic function without using technology. Make sure the \(x\)-intercepts, \(y\)-intercept, and vertex are plotted accurately.
Blank coordinate grid, origin O. X and y axis from negative 10 to 8, by 2s.
(From Unit 7, Lesson 22.)

Problem 6

Here are two quadratic functions: \(f(x) = (x + 5)^2 + \frac12\) and \(g(x) = (x + 5)^2 + 1\).

Andre says that both \(f\) and \(g\) have a minimum value, and that the minimum value of \(f\) is less than that of \(g\). Do you agree? Explain your reasoning.

(From Unit 7, Lesson 23.)

Problem 7

Function \(p\) is defined by the equation \(p(x)=(x + 10)^2 - 3\).

Function \(q\) is represented by this graph.

Which function has the smaller minimum? Explain your reasoning.

Function q on a grid. X axis from negative 2 to 10, by 2’s. Y axis from negative 4 to 20, by 4’s. Origin, O. Parabola opens upward with vertex around 5 comma 7.
(From Unit 7, Lesson 23.)

Problem 8

Without using graphing technology, sketch a graph that represents each quadratic function. Make sure the \(x\)-intercepts, \(y\)-intercept, and vertex are plotted accurately.

\(f(x) = x^2 + 4x + 3\)

Blank coordinate grid, origin O. X and y axis from negative 8 to 6, by 2s.

\(g(x)=x^2-4x+3\)

Blank coordinate grid, origin O. X and y axis from negative 8 to 6, by 2s.

\(h(x) = x^2 - 11x + 28\)

Blank coordinate grid, origin O. X and y axis from negative 8 to 6, by 2s.

(From Unit 7, Lesson 22.)