Algebra Examples

$x = - \frac{1}{2} y^{2} - 4 y - 7$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Combine $y^{2}$ and $\frac{1}{2}$ .

$x = - \frac{y^{2}}{2} - 4 y - 7$

Step 1.2

Complete the square for $- \frac{y^{2}}{2} - 4 y - 7$ .

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Step 1.2.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - \frac{1}{2}$

$b = - 4$

$c = - 7$

Step 1.2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 4}{2 (- \frac{1}{2})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $- 4$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $- 4$ .

$d = \frac{2 \cdot - 2}{2 (- \frac{1}{2})}$

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

Cancel the common factor.

$d = \frac{2 \cdot - 2}{2 (- \frac{1}{2})}$

Step 1.2.3.2.1.2.2

Rewrite the expression.

$d = \frac{- 2}{- \frac{1}{2}}$

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = - 2 (- 1 \cdot 2)$

Step 1.2.3.2.3

Multiply $- 2 (- 1 \cdot 2)$ .

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Step 1.2.3.2.3.1

Multiply $- 1$ by $2$ .

$d = - 2 \cdot - 2$

Step 1.2.3.2.3.2

Multiply $- 2$ by $- 2$ .

$d = 4$

Step 1.2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = - 7 - \frac{{(- 4)}^{2}}{4 (- \frac{1}{2})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

Cancel the common factor of ${(- 4)}^{2}$ and $4$ .

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Step 1.2.4.2.1.1.1

Rewrite $- 4$ as $- 1 (4)$ .

$e = - 7 - \frac{{(- 1 (4))}^{2}}{4 (- \frac{1}{2})}$

Step 1.2.4.2.1.1.2

Apply the product rule to $- 1 (4)$ .

$e = - 7 - \frac{{(- 1)}^{2} \cdot 4^{2}}{4 (- \frac{1}{2})}$

Step 1.2.4.2.1.1.3

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

$e = - 7 - \frac{1 \cdot 4^{2}}{4 (- \frac{1}{2})}$

Step 1.2.4.2.1.1.4

Multiply $4^{2}$ by $1$ .

$e = - 7 - \frac{4^{2}}{4 (- \frac{1}{2})}$

Step 1.2.4.2.1.1.5

Factor $4$ out of $4^{2}$ .

$e = - 7 - \frac{4 \cdot 4}{4 (- \frac{1}{2})}$

Step 1.2.4.2.1.1.6

Cancel the common factors.

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Step 1.2.4.2.1.1.6.1

Cancel the common factor.

$e = - 7 - \frac{4 \cdot 4}{4 (- \frac{1}{2})}$

Step 1.2.4.2.1.1.6.2

Rewrite the expression.

$e = - 7 - \frac{4}{- \frac{1}{2}}$

Step 1.2.4.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$e = - 7 - (4 (- 1 \cdot 2))$

Step 1.2.4.2.1.3

Multiply $- (4 (- 1 \cdot 2))$ .

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Step 1.2.4.2.1.3.1

Multiply $- 1$ by $2$ .

$e = - 7 - (4 \cdot - 2)$

Step 1.2.4.2.1.3.2

Multiply $4$ by $- 2$ .

$e = - 7 - - 8$

Step 1.2.4.2.1.3.3

Multiply $- 1$ by $- 8$ .

$e = - 7 + 8$

Step 1.2.4.2.2

Add $- 7$ and $8$ .

$e = 1$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{2} {(y + 4)}^{2} + 1$ .

$- \frac{1}{2} {(y + 4)}^{2} + 1$

Step 1.3

Set $x$ equal to the new right side.

$x = - \frac{1}{2} \cdot {(y + 4)}^{2} + 1$

Step 2

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

$a = - \frac{1}{2}$

$h = 1$

$k = - 4$

Step 3

Since the value of $a$ is negative, the parabola opens left.

Opens Left

Step 4

Find the vertex $(h, k)$ .

$(1, - 4)$

Step 5

Find $p$ , the distance from the vertex to the focus.

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Step 5.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 5.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot (- \frac{1}{2})}$

Step 5.3

Simplify.

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Step 5.3.1

Cancel the common factor of $1$ and $- 1$ .

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Step 5.3.1.1

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

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{2})}$

Step 5.3.1.2

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{2})}$

Step 5.3.2

Combine $4$ and $\frac{1}{2}$ .

$- \frac{1}{\frac{4}{2}}$

Step 5.3.3

Divide $4$ by $2$ .

$- \frac{1}{2}$

Step 6

Find the focus.

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Step 6.1

The focus of a parabola can be found by adding $p$ to the x-coordinate $h$ if the parabola opens left or right.

$(h + p, k)$

Step 6.2

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(\frac{1}{2}, - 4)$

Step 7

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

$y = - 4$

Step 8