A basketball player shoots a ball towards the hoop.

Question

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
4.04
1
2
3

Answer 1

To solve the problem, we will first identify the vertex of the quadratic function \( f(x) = -(x + 1.4)^2 + 6 \) and then calculate the function values for the given \( x \) values.

Part A: Identify the vertex

The vertex form of a quadratic function \( f(x) = a(x - h)^2 + k \) gives the vertex as the point \( (h, k) \).

In the function given: \[ f(x) = -(x + 1.4)^2 + 6 \] we can see that:

• \( h = -1.4 \)
• \( k = 6 \)

Therefore, the vertex is: \[ \text{Vertex} = (-1.4, 6) \]

Part B: Complete the table

Now we will calculate \( f(x) \) for the given \( x \) values.

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

5. For \( x = 4.04 \): \[ f(4.04) = - (4.04 + 1.4)^2 + 6 = - (5.44)^2 + 6 = -29.5936 + 6 = -23.5936 \]

Now we can compile these results into the completed table:

| x | f(x) | |-------|----------| | 0 | 4.04 | | 4.04 | -23.5936 | | 1 | 0.24 | | 2 | -5.56 | | 3 | -13.36 |

Summary

• Vertex: \((-1.4, 6)\)
• Completed Table:

| x | f(x) | |-------|----------| | 0 | 4.04 | | 4.04 | -23.5936 | | 1 | 0.24 | | 2 | -5.56 | | 3 | -13.36 |