Find the Maximum/Minimum Value f(x)=4(x+2)^2-6

$f (x) = 4 {(x + 2)}^{2} - 6$

Step 1

Simplify each term.

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$f (x) = 4 ((x + 2) (x + 2)) - 6$

Expand $(x + 2) (x + 2)$ using the FOIL Method.

Apply the distributive property.

$f (x) = 4 (x (x + 2) + 2 (x + 2)) - 6$

Apply the distributive property.

$f (x) = 4 (x \cdot x + x \cdot 2 + 2 (x + 2)) - 6$

Apply the distributive property.

$f (x) = 4 (x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2) - 6$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x) = 4 (x^{2} + x \cdot 2 + 2 x + 2 \cdot 2) - 6$

Move $2$ to the left of $x$ .

$f (x) = 4 (x^{2} + 2 \cdot x + 2 x + 2 \cdot 2) - 6$

Multiply $2$ by $2$ .

$f (x) = 4 (x^{2} + 2 x + 2 x + 4) - 6$

Add $2 x$ and $2 x$ .

$f (x) = 4 (x^{2} + 4 x + 4) - 6$

Apply the distributive property.

$f (x) = 4 x^{2} + 4 (4 x) + 4 \cdot 4 - 6$

Simplify.

Multiply $4$ by $4$ .

$f (x) = 4 x^{2} + 16 x + 4 \cdot 4 - 6$

Multiply $4$ by $4$ .

$f (x) = 4 x^{2} + 16 x + 16 - 6$

Step 2

Subtract $6$ from $16$ .

$f (x) = 4 x^{2} + 16 x + 10$

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 $b$ .

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

Remove parentheses.

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

Simplify $- \frac{16}{2 (4)}$ .

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

Factor $2$ out of $16$ .

$x = - \frac{2 \cdot 8}{2 \cdot 4}$

Cancel the common factors.

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

$x = - \frac{2 \cdot 8}{2 (4)}$

Cancel the common factor.

$x = - \frac{2 \cdot 8}{2 \cdot 4}$

Rewrite the expression.

$x = - \frac{8}{4}$

Cancel the common factor of $8$ and $4$ .

Factor $4$ out of $8$ .

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

Cancel the common factors.

Factor $4$ out of $4$ .

$x = - \frac{4 \cdot 2}{4 (1)}$

Cancel the common factor.

$x = - \frac{4 \cdot 2}{4 \cdot 1}$

Rewrite the expression.

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

Divide $2$ by $1$ .

$x = - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$x = - 2$

Step 5

Evaluate $f (- 2)$ .

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = 4 {(- 2)}^{2} + 16 (- 2) + 10$

Simplify the result.

Simplify each term.

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

$f (- 2) = 4 \cdot 4 + 16 (- 2) + 10$

Multiply $4$ by $4$ .

$f (- 2) = 16 + 16 (- 2) + 10$

Multiply $16$ by $- 2$ .

$f (- 2) = 16 - 32 + 10$

Simplify by adding and subtracting.

Subtract $32$ from $16$ .

$f (- 2) = - 16 + 10$

Add $- 16$ and $10$ .

$f (- 2) = - 6$

The final answer is $- 6$ .

$- 6$

Step 6

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

$(- 2, - 6)$

Step 7

Find the Maximum/Minimum Value f(x)=4(x+2)^2-6

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