Precalculus Examples

$y^{2} - 10 y + 4 x + 25 = 0$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

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Step 1.1.1

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

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Step 1.1.1.1

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

$- 10 y + 4 x + 25 = - y^{2}$

Step 1.1.1.2

Add $10 y$ to both sides of the equation.

$4 x + 25 = - y^{2} + 10 y$

Step 1.1.1.3

Subtract $25$ from both sides of the equation.

$4 x = - y^{2} + 10 y - 25$

Step 1.1.2

Divide each term in $4 x = - y^{2} + 10 y - 25$ by $4$ and simplify.

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Step 1.1.2.1

Divide each term in $4 x = - y^{2} + 10 y - 25$ by $4$ .

$\frac{4 x}{4} = \frac{- y^{2}}{4} + \frac{10 y}{4} + \frac{- 25}{4}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $4$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- y^{2}}{4} + \frac{10 y}{4} + \frac{- 25}{4}$

Step 1.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- y^{2}}{4} + \frac{10 y}{4} + \frac{- 25}{4}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Move the negative in front of the fraction.

$x = - \frac{y^{2}}{4} + \frac{10 y}{4} + \frac{- 25}{4}$

Step 1.1.2.3.1.2

Cancel the common factor of $10$ and $4$ .

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Step 1.1.2.3.1.2.1

Factor $2$ out of $10 y$ .

$x = - \frac{y^{2}}{4} + \frac{2 (5 y)}{4} + \frac{- 25}{4}$

Step 1.1.2.3.1.2.2

Cancel the common factors.

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Step 1.1.2.3.1.2.2.1

Factor $2$ out of $4$ .

$x = - \frac{y^{2}}{4} + \frac{2 (5 y)}{2 (2)} + \frac{- 25}{4}$

Step 1.1.2.3.1.2.2.2

Cancel the common factor.

$x = - \frac{y^{2}}{4} + \frac{2 (5 y)}{2 \cdot 2} + \frac{- 25}{4}$

Step 1.1.2.3.1.2.2.3

Rewrite the expression.

$x = - \frac{y^{2}}{4} + \frac{5 y}{2} + \frac{- 25}{4}$

Step 1.1.2.3.1.3

Move the negative in front of the fraction.

$x = - \frac{y^{2}}{4} + \frac{5 y}{2} - \frac{25}{4}$

Step 1.2

Complete the square for $- \frac{y^{2}}{4} + \frac{5 y}{2} - \frac{25}{4}$ .

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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 $c$ .

$a = - \frac{1}{4}$

$b = \frac{5}{2}$

$c = - \frac{25}{4}$

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

$d = \frac{\frac{5}{2}}{2 (- \frac{1}{4})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{5}{2} \cdot \frac{1}{2 (- \frac{1}{4})}$

Step 1.2.3.2.2

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

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Step 1.2.3.2.2.1

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

$d = \frac{5}{2} \cdot \frac{- 1 (- 1)}{2 (- \frac{1}{4})}$

Step 1.2.3.2.2.2

Move the negative in front of the fraction.

$d = \frac{5}{2} (- \frac{1}{2 (\frac{1}{4})})$

Step 1.2.3.2.3

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

$d = \frac{5}{2} (- \frac{1}{\frac{2}{4}})$

Step 1.2.3.2.4

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

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Step 1.2.3.2.4.1

Factor $2$ out of $2$ .

$d = \frac{5}{2} (- \frac{1}{\frac{2 (1)}{4}})$

Step 1.2.3.2.4.2

Cancel the common factors.

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Step 1.2.3.2.4.2.1

Factor $2$ out of $4$ .

$d = \frac{5}{2} (- \frac{1}{\frac{2 \cdot 1}{2 \cdot 2}})$

Step 1.2.3.2.4.2.2

Cancel the common factor.

$d = \frac{5}{2} (- \frac{1}{\frac{2 \cdot 1}{2 \cdot 2}})$

Step 1.2.3.2.4.2.3

Rewrite the expression.

$d = \frac{5}{2} (- \frac{1}{\frac{1}{2}})$

Step 1.2.3.2.5

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{5}{2} (- (1 \cdot 2))$

Step 1.2.3.2.6

Cancel the common factor of $2$ .

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Step 1.2.3.2.6.1

Factor $2$ out of $- (1 \cdot 2)$ .

$d = \frac{5}{2} (2 (- 1 \cdot (1)))$

Step 1.2.3.2.6.2

Cancel the common factor.

$d = \frac{5}{2} (2 (- 1 \cdot (1)))$

Step 1.2.3.2.6.3

Rewrite the expression.

$d = 5 (- 1 \cdot (1))$

Step 1.2.3.2.7

Multiply $- 1$ by $1$ .

$d = 5 \cdot - 1$

Step 1.2.3.2.8

Multiply $5$ by $- 1$ .

$d = - 5$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = - \frac{25}{4} - \frac{{(\frac{5}{2})}^{2}}{4 (- \frac{1}{4})}$

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

Simplify the numerator.

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Step 1.2.4.2.1.1.1

Apply the product rule to $\frac{5}{2}$ .

$e = - \frac{25}{4} - \frac{\frac{5^{2}}{2^{2}}}{4 (- \frac{1}{4})}$

Step 1.2.4.2.1.1.2

Raise $5$ to the power of $2$ .

$e = - \frac{25}{4} - \frac{\frac{25}{2^{2}}}{4 (- \frac{1}{4})}$

Step 1.2.4.2.1.1.3

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

$e = - \frac{25}{4} - \frac{\frac{25}{4}}{4 (- \frac{1}{4})}$

Step 1.2.4.2.1.2

Simplify the denominator.

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Step 1.2.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = - \frac{25}{4} - \frac{\frac{25}{4}}{- 4 (\frac{1}{4})}$

Step 1.2.4.2.1.2.2

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

$e = - \frac{25}{4} - \frac{\frac{25}{4}}{\frac{- 4}{4}}$

Step 1.2.4.2.1.3

Divide $- 4$ by $4$ .

$e = - \frac{25}{4} - \frac{\frac{25}{4}}{- 1}$

Step 1.2.4.2.1.4

Move the negative one from the denominator of $\frac{\frac{25}{4}}{- 1}$ .

$e = - \frac{25}{4} - (- 1 \cdot \frac{25}{4})$

Step 1.2.4.2.1.5

Rewrite $- 1 \cdot \frac{25}{4}$ as $- \frac{25}{4}$ .

$e = - \frac{25}{4} - - \frac{25}{4}$

Step 1.2.4.2.1.6

Multiply $- - \frac{25}{4}$ .

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Step 1.2.4.2.1.6.1

Multiply $- 1$ by $- 1$ .

$e = - \frac{25}{4} + 1 (\frac{25}{4})$

Step 1.2.4.2.1.6.2

Multiply $\frac{25}{4}$ by $1$ .

$e = - \frac{25}{4} + \frac{25}{4}$

Step 1.2.4.2.2

Combine the numerators over the common denominator.

$e = \frac{- 25 + 25}{4}$

Step 1.2.4.2.3

Add $- 25$ and $25$ .

$e = \frac{0}{4}$

Step 1.2.4.2.4

Divide $0$ by $4$ .

$e = 0$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{4} {(y - 5)}^{2} + 0$ .

$- \frac{1}{4} {(y - 5)}^{2} + 0$

Step 1.3

Set $x$ equal to the new right side.

$x = - \frac{1}{4} \cdot {(y - 5)}^{2} + 0$

Step 2

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

$a = - \frac{1}{4}$

$h = 0$

$k = 5$

Step 3

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

Opens Left

Step 4

Find the vertex $(h, k)$ .

$(0, 5)$

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.

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

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 $- 1 (- 1)$ .

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

Step 5.3.1.2

Move the negative in front of the fraction.

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

Step 5.3.2

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

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

Step 5.3.3

Divide $4$ by $4$ .

$- \frac{1}{1}$

Step 5.3.4

Cancel the common factor of $1$ .

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Step 5.3.4.1

Cancel the common factor.

$- \frac{1}{1}$

Step 5.3.4.2

Rewrite the expression.

$- 1 \cdot 1$

Step 5.3.5

Multiply $- 1$ by $1$ .

$- 1$

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.

$(- 1, 5)$

Step 7

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

$y = 5$

Step 8

Find the directrix.

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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.

$x = 1$

Step 9

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

Direction: Opens Left

Vertex: $(0, 5)$

Focus: $(- 1, 5)$

Axis of Symmetry: $y = 5$

Directrix: $x = 1$

Step 10