Find the Maximum/Minimum Value y=0.005(x-60)^2+8

$y = 0.005 {(x - 60)}^{2} + 8$

Step 1

Simplify each term.

Rewrite ${(x - 60)}^{2}$ as $(x - 60) (x - 60)$ .

$f (x) = 0.005 ((x - 60) (x - 60)) + 8$

Expand $(x - 60) (x - 60)$ using the FOIL Method.

Apply the distributive property.

$f (x) = 0.005 (x (x - 60) - 60 (x - 60)) + 8$

Apply the distributive property.

$f (x) = 0.005 (x \cdot x + x \cdot - 60 - 60 (x - 60)) + 8$

Apply the distributive property.

$f (x) = 0.005 (x \cdot x + x \cdot - 60 - 60 x - 60 \cdot - 60) + 8$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x) = 0.005 (x^{2} + x \cdot - 60 - 60 x - 60 \cdot - 60) + 8$

Move $- 60$ to the left of $x$ .

$f (x) = 0.005 (x^{2} - 60 \cdot x - 60 x - 60 \cdot - 60) + 8$

Multiply $- 60$ by $- 60$ .

$f (x) = 0.005 (x^{2} - 60 x - 60 x + 3600) + 8$

Subtract $60 x$ from $- 60 x$ .

$f (x) = 0.005 (x^{2} - 120 x + 3600) + 8$

Apply the distributive property.

$f (x) = 0.005 x^{2} + 0.005 (- 120 x) + 0.005 \cdot 3600 + 8$

Simplify.

Multiply $- 120$ by $0.005$ .

$f (x) = 0.005 x^{2} - 0.6 x + 0.005 \cdot 3600 + 8$

Multiply $0.005$ by $3600$ .

$f (x) = 0.005 x^{2} - 0.6 x + 18 + 8$

Step 2

Add $18$ and $8$ .

$f (x) = 0.005 x^{2} - 0.6 x + 26$

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 $b$ .

$x = - \frac{- 0.6}{2 (0.005)}$

Remove parentheses.

$x = - \frac{- 0.6}{2 (0.005)}$

Simplify $- \frac{- 0.6}{2 (0.005)}$ .

Multiply $2$ by $0.005$ .

$x = - \frac{- 0.6}{0.01}$

Divide $- 0.6$ by $0.01$ .

$x = - - 60$

Multiply $- 1$ by $- 60$ .

$x = 60$

Step 5

Evaluate $f (60)$ .

Replace the variable $x$ with $60$ in the expression.

$f (60) = 0.005 {(60)}^{2} - 0.6 \cdot 60 + 26$

Simplify the result.

Simplify each term.

Raise $60$ to the power of $2$ .

$f (60) = 0.005 \cdot 3600 - 0.6 \cdot 60 + 26$

Multiply $0.005$ by $3600$ .

$f (60) = 18 - 0.6 \cdot 60 + 26$

Multiply $- 0.6$ by $60$ .

$f (60) = 18 - 36 + 26$

Simplify by adding and subtracting.

Subtract $36$ from $18$ .

$f (60) = - 18 + 26$

Add $- 18$ and $26$ .

$f (60) = 8$

The final answer is $8$ .

$8$

Step 6

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

$(60, 8)$

Step 7

Find the Maximum/Minimum Value y=0.005(x-60)^2+8

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