Precalculus Examples

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

Step 1

Write $f (x) = - 2 {(x + 3)}^{2} + 1$ as an equation.

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

Step 2

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 3

Subtract $1$ from both sides of the equation.

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

Step 4

Divide each term in $- 2 {(x + 3)}^{2} = y - 1$ by $- 2$ and simplify.

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Step 4.1

Divide each term in $- 2 {(x + 3)}^{2} = y - 1$ by $- 2$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 2$ .

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Divide ${(x + 3)}^{2}$ by $1$ .

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

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Move the negative in front of the fraction.

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

Step 4.3.1.2

Dividing two negative values results in a positive value.

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

Step 5

Factor.

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Step 5.1

Combine the numerators over the common denominator.

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

Step 5.2

Factor $- 1$ out of $- y$ .

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

Step 5.3

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 5.4

Factor $- 1$ out of $- (y) - 1 (- 1)$ .

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

Step 5.5

Move the negative in front of the fraction.

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

Step 5.6

Reorder terms.

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

Step 5.7

Remove parentheses.

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

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