Precalculus Examples

${(y + \frac{5}{2})}^{2} = - 5 (x - \frac{9}{2})$

Step 1

Isolate $x$ to the left side of the equation.

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Step 1.1

Rewrite the equation as $- 5 (x - \frac{9}{2}) = {(y + \frac{5}{2})}^{2}$ .

$- 5 (x - \frac{9}{2}) = {(y + \frac{5}{2})}^{2}$

Step 1.2

Divide each term in $- 5 (x - \frac{9}{2}) = {(y + \frac{5}{2})}^{2}$ by $- 5$ and simplify.

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Step 1.2.1

Divide each term in $- 5 (x - \frac{9}{2}) = {(y + \frac{5}{2})}^{2}$ by $- 5$ .

$\frac{- 5 (x - \frac{9}{2})}{- 5} = \frac{{(y + \frac{5}{2})}^{2}}{- 5}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 5$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 5 (x - \frac{9}{2})}{- 5} = \frac{{(y + \frac{5}{2})}^{2}}{- 5}$

Step 1.2.2.1.2

Divide $x - \frac{9}{2}$ by $1$ .

$x - \frac{9}{2} = \frac{{(y + \frac{5}{2})}^{2}}{- 5}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$x - \frac{9}{2} = - \frac{{(y + \frac{5}{2})}^{2}}{5}$

Step 1.3

Add $\frac{9}{2}$ to both sides of the equation.

$x = - \frac{{(y + \frac{5}{2})}^{2}}{5} + \frac{9}{2}$

Step 1.4

Reorder terms.

$x = - \frac{1}{5} \cdot {(y + \frac{5}{2})}^{2} + \frac{9}{2}$

Step 2

Use the vertex form, $x = a {(y - k)}^{2} + h$ , to determine the values of $a$ , $h$ , and $k$ .

$a = - \frac{1}{5}$

$h = \frac{9}{2}$

$k = - \frac{5}{2}$

Step 3

Find the vertex $(h, k)$ .

$(\frac{9}{2}, - \frac{5}{2})$

Step 4