Find the Maximum/Minimum Value y=3(x-3)^2-3

$y = 3 {(x - 3)}^{2} - 3$

Step 1

Simplify each term.

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

$f (x) = 3 ((x - 3) (x - 3)) - 3$

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

Apply the distributive property.

$f (x) = 3 (x (x - 3) - 3 (x - 3)) - 3$

Apply the distributive property.

$f (x) = 3 (x \cdot x + x \cdot - 3 - 3 (x - 3)) - 3$

Apply the distributive property.

$f (x) = 3 (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3) - 3$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

$f (x) = 3 (x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3) - 3$

Multiply $- 3$ by $- 3$ .

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

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

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

Apply the distributive property.

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

Simplify.

Multiply $- 6$ by $3$ .

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

Multiply $3$ by $9$ .

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

Step 2

Subtract $3$ from $27$ .

$f (x) = 3 x^{2} - 18 x + 24$

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 $b$ .

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

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 $- 9$ .

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

Cancel the common factors.

Factor $3$ out of $3$ .

$x = - \frac{3 \cdot - 3}{3 (1)}$

Cancel the common factor.

$x = - \frac{3 \cdot - 3}{3 \cdot 1}$

Rewrite the expression.

$x = - \frac{- 3}{1}$

Divide $- 3$ by $1$ .

$x = - - 3$

Multiply $- 1$ by $- 3$ .

$x = 3$

Step 5

Evaluate $f (3)$ .

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

$f (3) = 3 {(3)}^{2} - 18 \cdot 3 + 24$

Simplify the result.

Simplify each term.

Multiply $3$ by ${(3)}^{2}$ by adding the exponents.

Multiply $3$ by ${(3)}^{2}$ .

Raise $3$ to the power of $1$ .

$f (3) = 3 {(3)}^{2} - 18 \cdot 3 + 24$

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

$f (3) = 3^{1 + 2} - 18 \cdot 3 + 24$

Add $1$ and $2$ .

$f (3) = 3^{3} - 18 \cdot 3 + 24$

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

$f (3) = 27 - 18 \cdot 3 + 24$

Multiply $- 18$ by $3$ .

$f (3) = 27 - 54 + 24$

Simplify by adding and subtracting.

Subtract $54$ from $27$ .

$f (3) = - 27 + 24$

Add $- 27$ and $24$ .

$f (3) = - 3$

The final answer is $- 3$ .

$- 3$

Step 6

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

$(3, - 3)$

Step 7

Find the Maximum/Minimum Value y=3(x-3)^2-3

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