Find the Maximum/Minimum Value h(x)=-5(x-1)^2+45

$h (x) = - 5 {(x - 1)}^{2} + 45$

Step 1

Simplify each term.

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$h (x) = - 5 ((x - 1) (x - 1)) + 45$

Expand $(x - 1) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$h (x) = - 5 (x (x - 1) - 1 (x - 1)) + 45$

Apply the distributive property.

$h (x) = - 5 (x \cdot x + x \cdot - 1 - 1 (x - 1)) + 45$

Apply the distributive property.

$h (x) = - 5 (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1) + 45$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$h (x) = - 5 (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1) + 45$

Move $- 1$ to the left of $x$ .

$h (x) = - 5 (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1) + 45$

Rewrite $- 1 x$ as $- x$ .

$h (x) = - 5 (x^{2} - x - 1 x - 1 \cdot - 1) + 45$

Rewrite $- 1 x$ as $- x$ .

$h (x) = - 5 (x^{2} - x - x - 1 \cdot - 1) + 45$

Multiply $- 1$ by $- 1$ .

$h (x) = - 5 (x^{2} - x - x + 1) + 45$

Subtract $x$ from $- x$ .

$h (x) = - 5 (x^{2} - 2 x + 1) + 45$

Apply the distributive property.

$h (x) = - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1 + 45$

Simplify.

Multiply $- 2$ by $- 5$ .

$h (x) = - 5 x^{2} + 10 x - 5 \cdot 1 + 45$

Multiply $- 5$ by $1$ .

$h (x) = - 5 x^{2} + 10 x - 5 + 45$

Step 2

Add $- 5$ and $45$ .

$h (x) = - 5 x^{2} + 10 x + 40$

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 $b$ .

$x = - \frac{10}{2 (- 5)}$

Remove parentheses.

$x = - \frac{10}{2 (- 5)}$

Simplify $- \frac{10}{2 (- 5)}$ .

Cancel the common factor of $10$ and $2$ .

Factor $2$ out of $10$ .

$x = - \frac{2 \cdot 5}{2 \cdot - 5}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 5$ .

$x = - \frac{2 \cdot 5}{2 (- 5)}$

Cancel the common factor.

$x = - \frac{2 \cdot 5}{2 \cdot - 5}$

Rewrite the expression.

$x = - \frac{5}{- 5}$

Cancel the common factor of $5$ and $- 5$ .

Factor $5$ out of $5$ .

$x = - \frac{5 (1)}{- 5}$

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

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

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

Multiply $- 1$ by $1$ .

$x = - - 1$

Multiply $- 1$ by $- 1$ .

$x = 1$

Step 5

Evaluate $f (1)$ .

Replace the variable $x$ with $1$ in the expression.

$h (1) = - 5 {(1)}^{2} + 10 (1) + 40$

Simplify the result.

Simplify each term.

One to any power is one.

$h (1) = - 5 \cdot 1 + 10 (1) + 40$

Multiply $- 5$ by $1$ .

$h (1) = - 5 + 10 (1) + 40$

Multiply $10$ by $1$ .

$h (1) = - 5 + 10 + 40$

Simplify by adding numbers.

Add $- 5$ and $10$ .

$h (1) = 5 + 40$

Add $5$ and $40$ .

$h (1) = 45$

The final answer is $45$ .

$45$

Step 6

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

$(1, 45)$

Step 7

Find the Maximum/Minimum Value h(x)=-5(x-1)^2+45

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