Algebra Examples

$- 3 {(y + 1)}^{2} + 7$

Step 1

Simplify each term.

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Step 1.1

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

$f (y) = - 3 ((y + 1) (y + 1)) + 7$

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

$f (y) = - 3 (y (y + 1) + 1 (y + 1)) + 7$

Step 1.2.2

Apply the distributive property.

$f (y) = - 3 (y \cdot y + y \cdot 1 + 1 (y + 1)) + 7$

Step 1.2.3

Apply the distributive property.

$f (y) = - 3 (y \cdot y + y \cdot 1 + 1 y + 1 \cdot 1) + 7$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $y$ by $y$ .

$f (y) = - 3 (y^{2} + y \cdot 1 + 1 y + 1 \cdot 1) + 7$

Step 1.3.1.2

Multiply $y$ by $1$ .

$f (y) = - 3 (y^{2} + y + 1 y + 1 \cdot 1) + 7$

Step 1.3.1.3

Multiply $y$ by $1$ .

$f (y) = - 3 (y^{2} + y + y + 1 \cdot 1) + 7$

Step 1.3.1.4

Multiply $1$ by $1$ .

$f (y) = - 3 (y^{2} + y + y + 1) + 7$

Step 1.3.2

Add $y$ and $y$ .

$f (y) = - 3 (y^{2} + 2 y + 1) + 7$

Step 1.4

Apply the distributive property.

$f (y) = - 3 y^{2} - 3 (2 y) - 3 \cdot 1 + 7$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $2$ by $- 3$ .

$f (y) = - 3 y^{2} - 6 y - 3 \cdot 1 + 7$

Step 1.5.2

Multiply $- 3$ by $1$ .

$f (y) = - 3 y^{2} - 6 y - 3 + 7$

Step 2

Add $- 3$ and $7$ .

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

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 $f (- \frac{b}{2 a})$ .

$f_{max}$ $y = a y^{2} + b y + c$ occurs at $y = - \frac{b}{2 a}$

Step 4

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

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Step 4.1

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

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

Step 4.2

Remove parentheses.

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

Step 4.3

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

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Step 4.3.1

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

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Step 4.3.1.1

Factor $2$ out of $- 6$ .

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

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

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

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

Step 4.3.1.2.2

Cancel the common factor.

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

Step 4.3.1.2.3

Rewrite the expression.

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

Step 4.3.2

Cancel the common factor of $- 3$ .

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Step 4.3.2.1

Cancel the common factor.

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

Step 4.3.2.2

Rewrite the expression.

$y = - 1 \cdot 1$

Step 4.3.3

Multiply $- 1$ by $1$ .

$y = - 1$

Step 5

Evaluate $f (- 1)$ .

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Step 5.1

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

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

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

$f (- 1) = - 3 \cdot 1 - 6 \cdot - 1 + 4$

Step 5.2.1.2

Multiply $- 3$ by $1$ .

$f (- 1) = - 3 - 6 \cdot - 1 + 4$

Step 5.2.1.3

Multiply $- 6$ by $- 1$ .

$f (- 1) = - 3 + 6 + 4$

Step 5.2.2

Simplify by adding numbers.

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Step 5.2.2.1

Add $- 3$ and $6$ .

$f (- 1) = 3 + 4$

Step 5.2.2.2

Add $3$ and $4$ .

$f (- 1) = 7$

Step 5.2.3

The final answer is $7$ .

$7$

Step 6

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

$(- 1, 7)$

Step 7