# 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.

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.

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 because its decimal never terminates or forms a repeating pattern. I also know that $$\frac29$$ is a rational number because its decimal forms a repeating pattern. 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.
(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.

(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.

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

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

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

(From Unit 7, Lesson 22.)