Question

A basketball player shoots a ball towards the hoop.

The trajectory of the ball can be modeled by the quadratic function f(x) = −(x+1.4)2+6
f
(
x
)

=

−
(
x
+
1.4
)
2
+
6
, where x represents the horizontal distance in feet from the player, and f(x) represents the height of the ball in feet.



(1 point)
Part A: Identify the vertex (write your answer as an ordered pair):

Part B: Complete the table

x f(x)
0
1
2
3

Answer 1

To analyze the given quadratic function \( f(x) = - (x + 1.4)^2 + 6 \), let's go through the parts step-by-step.

**Part A: Identify the vertex.**
The vertex form of a quadratic function is \( f(x) = a(x-h)^2 + k \), where \( (h, k) \) is the vertex. The function you've provided is:

\[
f(x) = - (x + 1.4)^2 + 6
\]

From this equation, we can identify:

- The term \( (x + 1.4) \) indicates that \( h = -1.4 \).
- The maximum value of the function occurs when \( x + 1.4 = 0 \), which corresponds to \( k = 6 \).

So, the vertex is:

\[
\text{Vertex} = (-1.4, 6)
\]

**Part B: Complete the table.**

To complete the table, we will calculate the values of \( f(x) \) for \( x = 0, 1, 2, 3 \).

1. For \( x = 0 \):
\[
f(0) = - (0 + 1.4)^2 + 6 = - (1.4)^2 + 6 = -1.96 + 6 = 4.04
\]

2. For \( x = 1 \):
\[
f(1) = - (1 + 1.4)^2 + 6 = - (2.4)^2 + 6 = -5.76 + 6 = 0.24
\]

3. For \( x = 2 \):
\[
f(2) = - (2 + 1.4)^2 + 6 = - (3.4)^2 + 6 = -11.56 + 6 = -5.56
\]

4. For \( x = 3 \):
\[
f(3) = - (3 + 1.4)^2 + 6 = - (4.4)^2 + 6 = -19.36 + 6 = -13.36
\]

Now we can complete the table:

| \(x\) | \(f(x)\) |
|:----:|:------:|
| 0 | 4.04 |
| 1 | 0.24 |
| 2 | -5.56 |
| 3 | -13.36 |

### Summary:
- **Part A:** Vertex = \((-1.4, 6)\)
- **Part B:**
| \(x\) | \(f(x)\) |
|:----:|:------:|
| 0 | 4.04 |
| 1 | 0.24 |
| 2 | -5.56 |
| 3 | -13.36 |