Amir stands on a balcony and throws a ball to his dog, who is at ground level. The ball's height (in meters above the ground), $$ x $$ seconds after Amir threw it, is modeled by $$ h(x)=-(x+1)(x-7) $$ What is the maximum height that the ball will reach? meters

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UpStudy AI · Accepted Answer

To find the maximum height that the ball will reach, we need to analyze the quadratic function given by

$$
h(x) = -(x + 1)(x - 7).
$$

### Step 1: Expand the function

First, let's expand the function to put it in standard quadratic form $$ h(x) = ax^2 + bx + c $$.

$$
h(x) = -((x + 1)(x - 7)) = -(x^2 - 7x + x - 7) = -x^2 + 6x + 7.
$$

### Step 2: Identify the coefficients

From the expanded form $$ h(x) = -x^2 + 6x + 7 $$, we can identify the coefficients:
- $$ a = -1 $$
- $$ b = 6 $$
- $$ c = 7 $$

### Step 3: Find the vertex

The maximum height of a quadratic function occurs at the vertex. The x-coordinate of the vertex can be found using the formula:

$$
x = -\frac{b}{2a}.
$$

Substituting the values of $$ a $$ and $$ b $$:

$$
x = -\frac{6}{2 \cdot -1} = \frac{6}{2} = 3.
$$

### Step 4: Calculate the maximum height

Now, we substitute $$ x = 3 $$ back into the height function $$ h(x) $$ to find the maximum height:

$$
h(3) = -3^2 + 6 \cdot 3 + 7.
$$

Calculating this step by step:

1. $$ -3^2 = -9 $$
2. $$ 6 \cdot 3 = 18 $$
3. Adding these values: $$ -9 + 18 + 7 = 16 $$.

Thus, the maximum height that the ball will reach is

$$
\boxed{16} \text{ meters}.
$$
The maximum height the ball will reach is 16 meters.

Amir stands on a balcony and throws a ball to his dog, who is at ground level. The ball's height (in meters above the ground), \( x \) seconds after Amir threw it, is modeled by \( h(x)=-(x+1)(x-7) \) What is the maximum height that the ball will reach? meters

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