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Let's break down the information given and solve the questions based on the function provided:

The equation modeling the basketball player's shot is:
\[
y = -\frac{1}{20}(x^2 - 19x + 48)
\]
where:
- \(x\) is the horizontal distance in feet from the player, and
- \(y\) is the height of the basketball in feet.

### (a) Does the player make the shot?
To answer this, we need to determine if the basketball reaches the hoop, which is located at the point \((19, 10)\). In other words, we want to check if the function reaches a height of 10 feet when \(x = 19\).

Substitute \(x = 19\) into the equation:
\[
y = -\frac{1}{20}((19)^2 - 19(19) + 48)
\]
\[
y = -\frac{1}{20}(361 - 361 + 48)
\]
\[
y = -\frac{1}{20}(48)
\]
\[
y = -2.4
\]
Thus, when \(x = 19\), the height of the ball is \(y = -2.4\) feet, which means the ball does not reach the height of the hoop. So, **the player does not make the shot**.

### (b) What was the maximum height of the basketball?
The maximum height occurs at the vertex of the parabola, and for a quadratic equation \(y = ax^2 + bx + c\), the \(x\)-coordinate of the vertex is given by:
\[
x = -\frac{b}{2a}
\]
Here, the equation is already factored, but we can expand it:
\[
y = -\frac{1}{20}(x^2 - 19x + 48)
\]
Comparing this with the standard quadratic form, \(a = -\frac{1}{20}\) and \(b = \frac{19}{20}\).

To find the \(x\)-coordinate of the vertex:
\[
x = -\frac{-19}{2(-\frac{1}{20})} = \frac{19}{2} = 9.5
\]
Now, substitute \(x = 9.5\) into the original equation to find the maximum height:
\[
y = -\frac{1}{20}((9.5)^2 - 19(9.5) + 48)
\]
\[
y = -\frac{1}{20}(90.25 - 180.5 + 48)
\]
\[
y = -\frac{1}{20}(-42.25)
\]
\[
y = 2.1125
\]
Thus, the maximum height of the basketball is approximately **2.11 feet**, which is quite low for a basketball shot.

---

Do you have any further questions or need more details? Here are five questions to expand the problem:

1. Can we determine the point where the ball would land based on this function?
2. What would the equation of the basketball shot look like if it needed to reach a height of 10 feet at \(x = 19\)?
3. How does changing the value of \(a\) (the coefficient of \(x^2\)) affect the trajectory of the ball?
4. What would happen to the vertex of the parabola if the player shot from a different starting point?
5. How could you model the same shot but from a different angle or distance?

**Tip:** The vertex formula \(x = -\frac{b}{2a}\) is crucial for finding the highest or lowest point of a parabola.

A professional basketball player's shot is modeled by the function y = -1/20(x^2 - 19x + 48), where x and y are measured in feet. Does the player make the shot, and what was the maximum height of the basketball?

Learn how to analyze a basketball shot modeled by the quadratic function y = -1/20(x^2 - 19x + 48). This problem explores the concept of quadratic functions and parabolas to determine if the player makes the shot and calculates the maximum height of the ball.

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