Precalculus Examples

$y^{2} + 4 y - 8 x + 1 = 0$

Step 1

Rewrite the equation in vertex form.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 1.1.1.1

Subtract $y^{2}$ from both sides of the equation.

$4 y - 8 x + 1 = - y^{2}$

Step 1.1.1.2

Subtract $4 y$ from both sides of the equation.

$- 8 x + 1 = - y^{2} - 4 y$

Step 1.1.1.3

Subtract $1$ from both sides of the equation.

$- 8 x = - y^{2} - 4 y - 1$

Step 1.1.2

Divide each term in $- 8 x = - y^{2} - 4 y - 1$ by $- 8$ and simplify.

Tap for more steps...

Step 1.1.2.1

Divide each term in $- 8 x = - y^{2} - 4 y - 1$ by $- 8$ .

$\frac{- 8 x}{- 8} = \frac{- y^{2}}{- 8} + \frac{- 4 y}{- 8} + \frac{- 1}{- 8}$

Step 1.1.2.2

Simplify the left side.

Tap for more steps...

Step 1.1.2.2.1

Cancel the common factor of $- 8$ .

Tap for more steps...

Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} = \frac{- y^{2}}{- 8} + \frac{- 4 y}{- 8} + \frac{- 1}{- 8}$

Step 1.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- y^{2}}{- 8} + \frac{- 4 y}{- 8} + \frac{- 1}{- 8}$

Step 1.1.2.3

Simplify the right side.

Tap for more steps...

Step 1.1.2.3.1

Simplify each term.

Tap for more steps...

Step 1.1.2.3.1.1

Dividing two negative values results in a positive value.

$x = \frac{y^{2}}{8} + \frac{- 4 y}{- 8} + \frac{- 1}{- 8}$

Step 1.1.2.3.1.2

Cancel the common factor of $- 4$ and $- 8$ .

Tap for more steps...

Step 1.1.2.3.1.2.1

Factor $- 4$ out of $- 4 y$ .

$x = \frac{y^{2}}{8} + \frac{- 4 (y)}{- 8} + \frac{- 1}{- 8}$

Step 1.1.2.3.1.2.2

Cancel the common factors.

Tap for more steps...

Step 1.1.2.3.1.2.2.1

Factor $- 4$ out of $- 8$ .

$x = \frac{y^{2}}{8} + \frac{- 4 (y)}{- 4 (2)} + \frac{- 1}{- 8}$

Step 1.1.2.3.1.2.2.2

Cancel the common factor.

$x = \frac{y^{2}}{8} + \frac{- 4 y}{- 4 \cdot 2} + \frac{- 1}{- 8}$

Step 1.1.2.3.1.2.2.3

Rewrite the expression.

$x = \frac{y^{2}}{8} + \frac{y}{2} + \frac{- 1}{- 8}$

Step 1.1.2.3.1.3

Dividing two negative values results in a positive value.

$x = \frac{y^{2}}{8} + \frac{y}{2} + \frac{1}{8}$

Step 1.2

Complete the square for $\frac{y^{2}}{8} + \frac{y}{2} + \frac{1}{8}$ .

Tap for more steps...

Step 1.2.1

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

$a = \frac{1}{8}$

$b = \frac{1}{2}$

$c = \frac{1}{8}$

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 $d = \frac{b}{2 a}$ .

Tap for more steps...

Step 1.2.3.1

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

$d = \frac{\frac{1}{2}}{2 (\frac{1}{8})}$

Step 1.2.3.2

Simplify the right side.

Tap for more steps...

Step 1.2.3.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{1}{2} \cdot \frac{1}{2 (\frac{1}{8})}$

Step 1.2.3.2.2

Combine $2$ and $\frac{1}{8}$ .

$d = \frac{1}{2} \cdot \frac{1}{\frac{2}{8}}$

Step 1.2.3.2.3

Cancel the common factor of $2$ and $8$ .

Tap for more steps...

Step 1.2.3.2.3.1

Factor $2$ out of $2$ .

$d = \frac{1}{2} \cdot \frac{1}{\frac{2 (1)}{8}}$

Step 1.2.3.2.3.2

Cancel the common factors.

Tap for more steps...

Step 1.2.3.2.3.2.1

Factor $2$ out of $8$ .

$d = \frac{1}{2} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 4}}$

Step 1.2.3.2.3.2.2

Cancel the common factor.

$d = \frac{1}{2} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 4}}$

Step 1.2.3.2.3.2.3

Rewrite the expression.

$d = \frac{1}{2} \cdot \frac{1}{\frac{1}{4}}$

Step 1.2.3.2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.3.2.5

Multiply $4$ by $1$ .

$d = \frac{1}{2} \cdot 4$

Step 1.2.3.2.6

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.3.2.6.1

Factor $2$ out of $4$ .

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

Step 1.2.3.2.6.2

Cancel the common factor.

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

Step 1.2.3.2.6.3

Rewrite the expression.

$d = 2$

Step 1.2.4

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

Tap for more steps...

Step 1.2.4.1

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

$e = \frac{1}{8} - \frac{{(\frac{1}{2})}^{2}}{4 (\frac{1}{8})}$

Step 1.2.4.2

Simplify the right side.

Tap for more steps...

Step 1.2.4.2.1

Simplify each term.

Tap for more steps...

Step 1.2.4.2.1.1

Simplify the numerator.

Tap for more steps...

Step 1.2.4.2.1.1.1

Apply the product rule to $\frac{1}{2}$ .

$e = \frac{1}{8} - \frac{\frac{1^{2}}{2^{2}}}{4 (\frac{1}{8})}$

Step 1.2.4.2.1.1.2

One to any power is one.

$e = \frac{1}{8} - \frac{\frac{1}{2^{2}}}{4 (\frac{1}{8})}$

Step 1.2.4.2.1.1.3

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

$e = \frac{1}{8} - \frac{\frac{1}{4}}{4 (\frac{1}{8})}$

Step 1.2.4.2.1.2

Combine $4$ and $\frac{1}{8}$ .

$e = \frac{1}{8} - \frac{\frac{1}{4}}{\frac{4}{8}}$

Step 1.2.4.2.1.3

Cancel the common factor of $4$ and $8$ .

Tap for more steps...

Step 1.2.4.2.1.3.1

Factor $4$ out of $4$ .

$e = \frac{1}{8} - \frac{\frac{1}{4}}{\frac{4 (1)}{8}}$

Step 1.2.4.2.1.3.2

Cancel the common factors.

Tap for more steps...

Step 1.2.4.2.1.3.2.1

Factor $4$ out of $8$ .

$e = \frac{1}{8} - \frac{\frac{1}{4}}{\frac{4 \cdot 1}{4 \cdot 2}}$

Step 1.2.4.2.1.3.2.2

Cancel the common factor.

$e = \frac{1}{8} - \frac{\frac{1}{4}}{\frac{4 \cdot 1}{4 \cdot 2}}$

Step 1.2.4.2.1.3.2.3

Rewrite the expression.

$e = \frac{1}{8} - \frac{\frac{1}{4}}{\frac{1}{2}}$

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{1}{8} - (\frac{1}{4} \cdot 2)$

Step 1.2.4.2.1.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.4.2.1.5.1

Factor $2$ out of $4$ .

$e = \frac{1}{8} - (\frac{1}{2 (2)} \cdot 2)$

Step 1.2.4.2.1.5.2

Cancel the common factor.

$e = \frac{1}{8} - (\frac{1}{2 \cdot 2} \cdot 2)$

Step 1.2.4.2.1.5.3

Rewrite the expression.

$e = \frac{1}{8} - \frac{1}{2}$

Step 1.2.4.2.2

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = \frac{1}{8} - \frac{1}{2} \cdot \frac{4}{4}$

Step 1.2.4.2.3

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 1.2.4.2.3.1

Multiply $\frac{1}{2}$ by $\frac{4}{4}$ .

$e = \frac{1}{8} - \frac{4}{2 \cdot 4}$

Step 1.2.4.2.3.2

Multiply $2$ by $4$ .

$e = \frac{1}{8} - \frac{4}{8}$

Step 1.2.4.2.4

Combine the numerators over the common denominator.

$e = \frac{1 - 4}{8}$

Step 1.2.4.2.5

Subtract $4$ from $1$ .

$e = \frac{- 3}{8}$

Step 1.2.4.2.6

Move the negative in front of the fraction.

$e = - \frac{3}{8}$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{8} {(y + 2)}^{2} - \frac{3}{8}$ .

$\frac{1}{8} {(y + 2)}^{2} - \frac{3}{8}$

Step 1.3

Set $x$ equal to the new right side.

$x = \frac{1}{8} \cdot {(y + 2)}^{2} - \frac{3}{8}$

Step 2

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

$a = \frac{1}{8}$

$h = - \frac{3}{8}$

$k = - 2$

Step 3

Since the value of $a$ is positive, the parabola opens right.

Opens Right

Step 4

Find the vertex $(h, k)$ .

$(- \frac{3}{8}, - 2)$

Step 5

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

Tap for more steps...

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.

$\frac{1}{4 \cdot \frac{1}{8}}$

Step 5.3

Simplify.

Tap for more steps...

Step 5.3.1

Combine $4$ and $\frac{1}{8}$ .

$\frac{1}{\frac{4}{8}}$

Step 5.3.2

Cancel the common factor of $4$ and $8$ .

Tap for more steps...

Step 5.3.2.1

Factor $4$ out of $4$ .

$\frac{1}{\frac{4 (1)}{8}}$

Step 5.3.2.2

Cancel the common factors.

Tap for more steps...

Step 5.3.2.2.1

Factor $4$ out of $8$ .

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

Step 5.3.2.2.2

Cancel the common factor.

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

Step 5.3.2.2.3

Rewrite the expression.

$\frac{1}{\frac{1}{2}}$

Step 5.3.3

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 2$

Step 5.3.4

Multiply $2$ by $1$ .

$2$

Step 6

Find the focus.

Tap for more steps...

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.

$(\frac{13}{8}, - 2)$

Step 7

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

$y = - 2$

Step 8

Find the directrix.

Tap for more steps...

Step 8.1

The directrix of a parabola is the vertical line found by subtracting $p$ from the x-coordinate $h$ of the vertex if the parabola opens left or right.

$x = h - p$

Step 8.2

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

$x = - \frac{19}{8}$

Step 9

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Right

Vertex: $(- \frac{3}{8}, - 2)$

Focus: $(\frac{13}{8}, - 2)$

Axis of Symmetry: $y = - 2$

Directrix: $x = - \frac{19}{8}$

Step 10