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 \(h(t)=\text5t^2+20t+18\). Height is measured in meters.
Answer each question without graphing. Explain or show your reasoning.
 After how many seconds does the object reach a height of 33 meters?
 When does the object reach its maximum height?
 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 fivefoot 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 \(h(x)=\text0.078x^2+0.7x+5.5\), 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."

Here is an argument that explains why \(\sqrt3 + \frac29\) is irrational. Complete the missing parts of the argument.
 Let \(x = \sqrt3 + \frac29\). If \(x\) were rational, then \(x  \frac29\) would also be rational because . . . .
 But \(x  \frac29\) is not rational because . . . .
 Since \(x\) is not rational, it must be . . . .
 Use the same type of argument to explain why \(\sqrt3 \boldcdot \frac29\) is irrational.
Problem 5
The following expressions all define the same quadratic function.
\(x^2+2x8\)
\((x+4)(x2)\)
\((x+1)^29\)
 What is the \(y\)intercept of the graph of the function?
 What are the \(x\)intercepts of the graph?
 What is the vertex of the graph?
 Sketch a graph of the quadratic function without using technology. Make sure the \(x\)intercepts, \(y\)intercept, and vertex are plotted accurately.
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.
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.
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.
\(f(x) = x^2 + 4x + 3\)
\(g(x)=x^24x+3\)
\(h(x) = x^2  11x + 28\)