Find the Maximum/Minimum Value y=-3(x+1)^2+4

$y = - 3 {(x + 1)}^{2} + 4$

Step 1

Simplify each term.

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

$f (x) = - 3 ((x + 1) (x + 1)) + 4$

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

Apply the distributive property.

$f (x) = - 3 (x (x + 1) + 1 (x + 1)) + 4$

Apply the distributive property.

$f (x) = - 3 (x \cdot x + x \cdot 1 + 1 (x + 1)) + 4$

Apply the distributive property.

$f (x) = - 3 (x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1) + 4$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x) = - 3 (x^{2} + x \cdot 1 + 1 x + 1 \cdot 1) + 4$

Multiply $x$ by $1$ .

$f (x) = - 3 (x^{2} + x + 1 x + 1 \cdot 1) + 4$

Multiply $x$ by $1$ .

$f (x) = - 3 (x^{2} + x + x + 1 \cdot 1) + 4$

Multiply $1$ by $1$ .

$f (x) = - 3 (x^{2} + x + x + 1) + 4$

Add $x$ and $x$ .

$f (x) = - 3 (x^{2} + 2 x + 1) + 4$

Apply the distributive property.

$f (x) = - 3 x^{2} - 3 (2 x) - 3 \cdot 1 + 4$

Simplify.

Multiply $2$ by $- 3$ .

$f (x) = - 3 x^{2} - 6 x - 3 \cdot 1 + 4$

Multiply $- 3$ by $1$ .

$f (x) = - 3 x^{2} - 6 x - 3 + 4$

Step 2

Add $- 3$ and $4$ .

$f (x) = - 3 x^{2} - 6 x + 1$

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 $b$ .

$x = - \frac{- 6}{2 (- 3)}$

Remove parentheses.

$x = - \frac{- 6}{2 (- 3)}$

Simplify $- \frac{- 6}{2 (- 3)}$ .

Cancel the common factor of $- 6$ and $2$ .

Factor $2$ out of $- 6$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $- 3$ .

Cancel the common factor.

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

Rewrite the expression.

$x = - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$x = - 1$

Step 5

Evaluate $f (- 1)$ .

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

$f (- 1) = - 3 {(- 1)}^{2} - 6 \cdot - 1 + 1$

Simplify the result.

Simplify each term.

Raise $- 1$ to the power of $2$ .

$f (- 1) = - 3 \cdot 1 - 6 \cdot - 1 + 1$

Multiply $- 3$ by $1$ .

$f (- 1) = - 3 - 6 \cdot - 1 + 1$

Multiply $- 6$ by $- 1$ .

$f (- 1) = - 3 + 6 + 1$

Simplify by adding numbers.

Add $- 3$ and $6$ .

$f (- 1) = 3 + 1$

Add $3$ and $1$ .

$f (- 1) = 4$

The final answer is $4$ .

$4$

Step 6

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

$(- 1, 4)$

Step 7

Find the Maximum/Minimum Value y=-3(x+1)^2+4

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