Find the Maximum/Minimum Value y=4x^2-32x+70

$y = 4 x^{2} - 32 x + 70$

Step 1

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 2

Find the value of $x = - \frac{b}{2 a}$ .

Substitute in the values of $a$ and $b$ .

$x = - \frac{- 32}{2 (4)}$

Remove parentheses.

$x = - \frac{- 32}{2 (4)}$

Simplify $- \frac{- 32}{2 (4)}$ .

Cancel the common factor of $- 32$ and $2$ .

Factor $2$ out of $- 32$ .

$x = - \frac{2 \cdot - 16}{2 \cdot 4}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 4$ .

$x = - \frac{2 \cdot - 16}{2 (4)}$

Cancel the common factor.

$x = - \frac{2 \cdot - 16}{2 \cdot 4}$

Rewrite the expression.

$x = - \frac{- 16}{4}$

Cancel the common factor of $- 16$ and $4$ .

Factor $4$ out of $- 16$ .

$x = - \frac{4 \cdot - 4}{4}$

Cancel the common factors.

Factor $4$ out of $4$ .

$x = - \frac{4 \cdot - 4}{4 (1)}$

Cancel the common factor.

$x = - \frac{4 \cdot - 4}{4 \cdot 1}$

Rewrite the expression.

$x = - \frac{- 4}{1}$

Divide $- 4$ by $1$ .

$x = - - 4$

Multiply $- 1$ by $- 4$ .

$x = 4$

Step 3

Evaluate $f (4)$ .

Replace the variable $x$ with $4$ in the expression.

$f (4) = 4 {(4)}^{2} - 32 \cdot 4 + 70$

Simplify the result.

Simplify each term.

Multiply $4$ by ${(4)}^{2}$ by adding the exponents.

Multiply $4$ by ${(4)}^{2}$ .

Raise $4$ to the power of $1$ .

$f (4) = 4 {(4)}^{2} - 32 \cdot 4 + 70$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (4) = 4^{1 + 2} - 32 \cdot 4 + 70$

Add $1$ and $2$ .

$f (4) = 4^{3} - 32 \cdot 4 + 70$

Raise $4$ to the power of $3$ .

$f (4) = 64 - 32 \cdot 4 + 70$

Multiply $- 32$ by $4$ .

$f (4) = 64 - 128 + 70$

Simplify by adding and subtracting.

Subtract $128$ from $64$ .

$f (4) = - 64 + 70$

Add $- 64$ and $70$ .

$f (4) = 6$

The final answer is $6$ .

$6$

Step 4

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

$(4, 6)$

Step 5

Find the Maximum/Minimum Value y=4x^2-32x+70

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