Find the Maximum/Minimum Value h(x)=-3(x-3)^2+108

$h (x) = - 3 {(x - 3)}^{2} + 108$

Step 1

Simplify each term.

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$h (x) = - 3 ((x - 3) (x - 3)) + 108$

Expand $(x - 3) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$h (x) = - 3 (x (x - 3) - 3 (x - 3)) + 108$

Apply the distributive property.

$h (x) = - 3 (x \cdot x + x \cdot - 3 - 3 (x - 3)) + 108$

Apply the distributive property.

$h (x) = - 3 (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3) + 108$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$h (x) = - 3 (x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3) + 108$

Move $- 3$ to the left of $x$ .

$h (x) = - 3 (x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3) + 108$

Multiply $- 3$ by $- 3$ .

$h (x) = - 3 (x^{2} - 3 x - 3 x + 9) + 108$

Subtract $3 x$ from $- 3 x$ .

$h (x) = - 3 (x^{2} - 6 x + 9) + 108$

Apply the distributive property.

$h (x) = - 3 x^{2} - 3 (- 6 x) - 3 \cdot 9 + 108$

Simplify.

Multiply $- 6$ by $- 3$ .

$h (x) = - 3 x^{2} + 18 x - 3 \cdot 9 + 108$

Multiply $- 3$ by $9$ .

$h (x) = - 3 x^{2} + 18 x - 27 + 108$

Step 2

Add $- 27$ and $108$ .

$h (x) = - 3 x^{2} + 18 x + 81$

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 $b$ .

$x = - \frac{18}{2 (- 3)}$

Remove parentheses.

$x = - \frac{18}{2 (- 3)}$

Simplify $- \frac{18}{2 (- 3)}$ .

Cancel the common factor of $18$ and $2$ .

Factor $2$ out of $18$ .

$x = - \frac{2 \cdot 9}{2 \cdot - 3}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 3$ .

$x = - \frac{2 \cdot 9}{2 (- 3)}$

Cancel the common factor.

$x = - \frac{2 \cdot 9}{2 \cdot - 3}$

Rewrite the expression.

$x = - \frac{9}{- 3}$

Cancel the common factor of $9$ and $- 3$ .

Factor $3$ out of $9$ .

$x = - \frac{3 (3)}{- 3}$

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 5

Evaluate $f (3)$ .

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

$h (3) = - 3 {(3)}^{2} + 18 (3) + 81$

Simplify the result.

Simplify each term.

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

$h (3) = - 3 \cdot 9 + 18 (3) + 81$

Multiply $- 3$ by $9$ .

$h (3) = - 27 + 18 (3) + 81$

Multiply $18$ by $3$ .

$h (3) = - 27 + 54 + 81$

Simplify by adding numbers.

Add $- 27$ and $54$ .

$h (3) = 27 + 81$

Add $27$ and $81$ .

$h (3) = 108$

The final answer is $108$ .

$108$

Step 6

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

$(3, 108)$

Step 7

Find the Maximum/Minimum Value h(x)=-3(x-3)^2+108

Related Questions

Support LCMGCF.com by disabling your adblocker.