Find the Maximum/Minimum Value y=(x-8)^2-5

$y = {(x - 8)}^{2} - 5$

Step 1

Simplify each term.

Rewrite ${(x - 8)}^{2}$ as $(x - 8) (x - 8)$ .

$f (x) = (x - 8) (x - 8) - 5$

Expand $(x - 8) (x - 8)$ using the FOIL Method.

Apply the distributive property.

$f (x) = x (x - 8) - 8 (x - 8) - 5$

Apply the distributive property.

$f (x) = x \cdot x + x \cdot - 8 - 8 (x - 8) - 5$

Apply the distributive property.

$f (x) = x \cdot x + x \cdot - 8 - 8 x - 8 \cdot - 8 - 5$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x) = x^{2} + x \cdot - 8 - 8 x - 8 \cdot - 8 - 5$

Move $- 8$ to the left of $x$ .

$f (x) = x^{2} - 8 \cdot x - 8 x - 8 \cdot - 8 - 5$

Multiply $- 8$ by $- 8$ .

$f (x) = x^{2} - 8 x - 8 x + 64 - 5$

Subtract $8 x$ from $- 8 x$ .

$f (x) = x^{2} - 16 x + 64 - 5$

Step 2

Subtract $5$ from $64$ .

$f (x) = x^{2} - 16 x + 59$

Step 3

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

$f_{min}$ $x = a x^{2} + b x + c$ 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 $b$ .

$x = - \frac{- 16}{2 (1)}$

Remove parentheses.

$x = - \frac{- 16}{2 (1)}$

Simplify $- \frac{- 16}{2 (1)}$ .

Cancel the common factor of $- 16$ and $2$ .

Factor $2$ out of $- 16$ .

$x = - \frac{2 \cdot - 8}{2 \cdot 1}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 1$ .

$x = - \frac{2 \cdot - 8}{2 (1)}$

Cancel the common factor.

$x = - \frac{2 \cdot - 8}{2 \cdot 1}$

Rewrite the expression.

$x = - \frac{- 8}{1}$

Divide $- 8$ by $1$ .

$x = - - 8$

Multiply $- 1$ by $- 8$ .

$x = 8$

Step 5

Evaluate $f (8)$ .

Replace the variable $x$ with $8$ in the expression.

$f (8) = {(8)}^{2} - 16 \cdot 8 + 59$

Simplify the result.

Simplify each term.

Raise $8$ to the power of $2$ .

$f (8) = 64 - 16 \cdot 8 + 59$

Multiply $- 16$ by $8$ .

$f (8) = 64 - 128 + 59$

Simplify by adding and subtracting.

Subtract $128$ from $64$ .

$f (8) = - 64 + 59$

Add $- 64$ and $59$ .

$f (8) = - 5$

The final answer is $- 5$ .

$- 5$

Step 6

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

$(8, - 5)$

Step 7

Find the Maximum/Minimum Value y=(x-8)^2-5

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