Lesson 24

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? 

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.

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.

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.

The following expressions all define the same quadratic function.

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.

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.

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

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

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

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