Find the Maximum/Minimum Value f(x)=-(x+7)^2+4

$f (x) = - {(x + 7)}^{2} + 4$

Step 1

Simplify each term.

Rewrite ${(x + 7)}^{2}$ as $(x + 7) (x + 7)$ .

$f (x) = - ((x + 7) (x + 7)) + 4$

Expand $(x + 7) (x + 7)$ using the FOIL Method.

Apply the distributive property.

$f (x) = - (x (x + 7) + 7 (x + 7)) + 4$

Apply the distributive property.

$f (x) = - (x \cdot x + x \cdot 7 + 7 (x + 7)) + 4$

Apply the distributive property.

$f (x) = - (x \cdot x + x \cdot 7 + 7 x + 7 \cdot 7) + 4$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x) = - (x^{2} + x \cdot 7 + 7 x + 7 \cdot 7) + 4$

Move $7$ to the left of $x$ .

$f (x) = - (x^{2} + 7 \cdot x + 7 x + 7 \cdot 7) + 4$

Multiply $7$ by $7$ .

$f (x) = - (x^{2} + 7 x + 7 x + 49) + 4$

Add $7 x$ and $7 x$ .

$f (x) = - (x^{2} + 14 x + 49) + 4$

Apply the distributive property.

$f (x) = - x^{2} - (14 x) - 1 \cdot 49 + 4$

Simplify.

Multiply $14$ by $- 1$ .

$f (x) = - x^{2} - 14 x - 1 \cdot 49 + 4$

Multiply $- 1$ by $49$ .

$f (x) = - x^{2} - 14 x - 49 + 4$

Step 2

Add $- 49$ and $4$ .

$f (x) = - x^{2} - 14 x - 45$

Step 3

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 4

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

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

$x = - \frac{- 14}{2 (- 1)}$

Remove parentheses.

$x = - \frac{- 14}{2 (- 1)}$

Simplify $- \frac{- 14}{2 (- 1)}$ .

Cancel the common factor of $- 14$ and $2$ .

Factor $2$ out of $- 14$ .

$x = - \frac{2 \cdot - 7}{2 \cdot - 1}$

Move the negative one from the denominator of $\frac{- 7}{- 1}$ .

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

Rewrite $- 1 \cdot - 7$ as $- - 7$ .

$x = - - - 7$

Multiply $- - - 7$ .

Multiply $- 1$ by $- 7$ .

$x = - 1 \cdot 7$

Multiply $- 1$ by $7$ .

$x = - 7$

Step 5

Evaluate $f (- 7)$ .

Replace the variable $x$ with $- 7$ in the expression.

$f (- 7) = - {(- 7)}^{2} - 14 \cdot - 7 - 45$

Simplify the result.

Simplify each term.

Raise $- 7$ to the power of $2$ .

$f (- 7) = - 1 \cdot 49 - 14 \cdot - 7 - 45$

Multiply $- 1$ by $49$ .

$f (- 7) = - 49 - 14 \cdot - 7 - 45$

Multiply $- 14$ by $- 7$ .

$f (- 7) = - 49 + 98 - 45$

Simplify by adding and subtracting.

Add $- 49$ and $98$ .

$f (- 7) = 49 - 45$

Subtract $45$ from $49$ .

$f (- 7) = 4$

The final answer is $4$ .

$4$

Step 6

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

$(- 7, 4)$

Step 7

Find the Maximum/Minimum Value f(x)=-(x+7)^2+4

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