Find the Maximum/Minimum Value f(x)=-(x-7)^2-26

$f (x) = - {(x - 7)}^{2} - 26$

Step 1

Simplify each term.

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

$f (x) = - ((x - 7) (x - 7)) - 26$

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

Apply the distributive property.

$f (x) = - (x (x - 7) - 7 (x - 7)) - 26$

Apply the distributive property.

$f (x) = - (x \cdot x + x \cdot - 7 - 7 (x - 7)) - 26$

Apply the distributive property.

$f (x) = - (x \cdot x + x \cdot - 7 - 7 x - 7 \cdot - 7) - 26$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x) = - (x^{2} + x \cdot - 7 - 7 x - 7 \cdot - 7) - 26$

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

$f (x) = - (x^{2} - 7 \cdot x - 7 x - 7 \cdot - 7) - 26$

Multiply $- 7$ by $- 7$ .

$f (x) = - (x^{2} - 7 x - 7 x + 49) - 26$

Subtract $7 x$ from $- 7 x$ .

$f (x) = - (x^{2} - 14 x + 49) - 26$

Apply the distributive property.

$f (x) = - x^{2} - (- 14 x) - 1 \cdot 49 - 26$

Simplify.

Multiply $- 14$ by $- 1$ .

$f (x) = - x^{2} + 14 x - 1 \cdot 49 - 26$

Multiply $- 1$ by $49$ .

$f (x) = - x^{2} + 14 x - 49 - 26$

Step 2

Subtract $26$ from $- 49$ .

$f (x) = - x^{2} + 14 x - 75$

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 $\frac{7}{- 1}$ .

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

Multiply $- (- 1 \cdot 7)$ .

Multiply $- 1$ by $7$ .

$x = - - 7$

Multiply $- 1$ by $- 7$ .

$x = 7$

Step 5

Evaluate $f (7)$ .

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

$f (7) = - {(7)}^{2} + 14 (7) - 75$

Simplify the result.

Simplify each term.

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

$f (7) = - 1 \cdot 49 + 14 (7) - 75$

Multiply $- 1$ by $49$ .

$f (7) = - 49 + 14 (7) - 75$

Multiply $14$ by $7$ .

$f (7) = - 49 + 98 - 75$

Simplify by adding and subtracting.

Add $- 49$ and $98$ .

$f (7) = 49 - 75$

Subtract $75$ from $49$ .

$f (7) = - 26$

The final answer is $- 26$ .

$- 26$

Step 6

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

$(7, - 26)$

Step 7

Find the Maximum/Minimum Value f(x)=-(x-7)^2-26

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