Find the Maximum/Minimum Value h(x)=-(x+1)(x-7)

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

Step 1

Simplify by multiplying through.

Apply the distributive property.

$h (x) = (- x - 1 \cdot 1) (x - 7)$

Multiply $- 1$ by $1$ .

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

Step 2

Expand $(- x - 1) (x - 7)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

$h (x) = - x \cdot x - x \cdot - 7 - 1 (x - 7)$

Apply the distributive property.

$h (x) = - x \cdot x - x \cdot - 7 - 1 x - 1 \cdot - 7$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$h (x) = - (x \cdot x) - x \cdot - 7 - 1 x - 1 \cdot - 7$

Multiply $x$ by $x$ .

$h (x) = - x^{2} - x \cdot - 7 - 1 x - 1 \cdot - 7$

Multiply $- 7$ by $- 1$ .

$h (x) = - x^{2} + 7 x - 1 x - 1 \cdot - 7$

Rewrite $- 1 x$ as $- x$ .

$h (x) = - x^{2} + 7 x - x - 1 \cdot - 7$

Multiply $- 1$ by $- 7$ .

$h (x) = - x^{2} + 7 x - x + 7$

Subtract $x$ from $7 x$ .

$h (x) = - x^{2} + 6 x + 7$

Step 4

The maximum of a quadratic function occurs at $x = - \frac{b}{2 a}$ . If $a$ is negative, the maximum value of the function is $f (- \frac{b}{2 a})$ .

$f_{max}$ $x = a x^{2} + b x + c$ occurs at $x = - \frac{b}{2 a}$

Step 5

Find the value of $x = - \frac{b}{2 a}$ .

Substitute in the values of $a$ and $b$ .

$x = - \frac{6}{2 (- 1)}$

Remove parentheses.

$x = - \frac{6}{2 (- 1)}$

Simplify $- \frac{6}{2 (- 1)}$ .

Cancel the common factor of $6$ and $2$ .

Factor $2$ out of $6$ .

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

Move the negative one from the denominator of $\frac{3}{- 1}$ .

$x = - (- 1 \cdot 3)$

Multiply $- (- 1 \cdot 3)$ .

Multiply $- 1$ by $3$ .

$x = - - 3$

Multiply $- 1$ by $- 3$ .

$x = 3$

Step 6

Evaluate $f (3)$ .

Replace the variable $x$ with $3$ in the expression.

$h (3) = - {(3)}^{2} + 6 (3) + 7$

Simplify the result.

Simplify each term.

Raise $3$ to the power of $2$ .

$h (3) = - 1 \cdot 9 + 6 (3) + 7$

Multiply $- 1$ by $9$ .

$h (3) = - 9 + 6 (3) + 7$

Multiply $6$ by $3$ .

$h (3) = - 9 + 18 + 7$

Simplify by adding numbers.

Add $- 9$ and $18$ .

$h (3) = 9 + 7$

Add $9$ and $7$ .

$h (3) = 16$

The final answer is $16$ .

$16$

Step 7

Use the $x$ and $y$ values to find where the maximum occurs.

$(3, 16)$

Step 8

Find the Maximum/Minimum Value h(x)=-(x+1)(x-7)

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