Precalculus Examples

$y^{2} + 6 y + 8 x + 25 = 0$

Step 1

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

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Step 1.1

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

$6 y + 8 x + 25 = - y^{2}$

Step 1.2

Subtract $6 y$ from both sides of the equation.

$8 x + 25 = - y^{2} - 6 y$

Step 1.3

Subtract $25$ from both sides of the equation.

$8 x = - y^{2} - 6 y - 25$

Step 2

Divide each term in $8 x = - y^{2} - 6 y - 25$ by $8$ and simplify.

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Step 2.1

Divide each term in $8 x = - y^{2} - 6 y - 25$ by $8$ .

$\frac{8 x}{8} = \frac{- y^{2}}{8} + \frac{- 6 y}{8} + \frac{- 25}{8}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $8$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{- y^{2}}{8} + \frac{- 6 y}{8} + \frac{- 25}{8}$

Step 2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- y^{2}}{8} + \frac{- 6 y}{8} + \frac{- 25}{8}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Move the negative in front of the fraction.

$x = - \frac{y^{2}}{8} + \frac{- 6 y}{8} + \frac{- 25}{8}$

Step 2.3.1.2

Cancel the common factor of $- 6$ and $8$ .

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Step 2.3.1.2.1

Factor $2$ out of $- 6 y$ .

$x = - \frac{y^{2}}{8} + \frac{2 (- 3 y)}{8} + \frac{- 25}{8}$

Step 2.3.1.2.2

Cancel the common factors.

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Step 2.3.1.2.2.1

Factor $2$ out of $8$ .

$x = - \frac{y^{2}}{8} + \frac{2 (- 3 y)}{2 (4)} + \frac{- 25}{8}$

Step 2.3.1.2.2.2

Cancel the common factor.

$x = - \frac{y^{2}}{8} + \frac{2 (- 3 y)}{2 \cdot 4} + \frac{- 25}{8}$

Step 2.3.1.2.2.3

Rewrite the expression.

$x = - \frac{y^{2}}{8} + \frac{- 3 y}{4} + \frac{- 25}{8}$

Step 2.3.1.3

Move the negative in front of the fraction.

$x = - \frac{y^{2}}{8} - \frac{3 y}{4} + \frac{- 25}{8}$

Step 2.3.1.4

Move the negative in front of the fraction.

$x = - \frac{y^{2}}{8} - \frac{3 y}{4} - \frac{25}{8}$

Step 3

Find the properties of the given parabola.

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Step 3.1

Rewrite the equation in vertex form.

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Step 3.1.1

Complete the square for $- \frac{y^{2}}{8} - \frac{3 y}{4} - \frac{25}{8}$ .

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Step 3.1.1.1

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

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

$b = - \frac{3}{4}$

$c = - \frac{25}{8}$

Step 3.1.1.2

Consider the vertex form of a parabola.

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

Step 3.1.1.3

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

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Step 3.1.1.3.1

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

$d = \frac{- \frac{3}{4}}{2 (- \frac{1}{8})}$

Step 3.1.1.3.2

Simplify the right side.

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Step 3.1.1.3.2.1

Dividing two negative values results in a positive value.

$d = \frac{\frac{3}{4}}{2 (\frac{1}{8})}$

Step 3.1.1.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{3}{4} \cdot \frac{1}{2 (\frac{1}{8})}$

Step 3.1.1.3.2.3

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

$d = \frac{3}{4} \cdot \frac{1}{\frac{2}{8}}$

Step 3.1.1.3.2.4

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

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Step 3.1.1.3.2.4.1

Factor $2$ out of $2$ .

$d = \frac{3}{4} \cdot \frac{1}{\frac{2 (1)}{8}}$

Step 3.1.1.3.2.4.2

Cancel the common factors.

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Step 3.1.1.3.2.4.2.1

Factor $2$ out of $8$ .

$d = \frac{3}{4} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 4}}$

Step 3.1.1.3.2.4.2.2

Cancel the common factor.

$d = \frac{3}{4} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 4}}$

Step 3.1.1.3.2.4.2.3

Rewrite the expression.

$d = \frac{3}{4} \cdot \frac{1}{\frac{1}{4}}$

Step 3.1.1.3.2.5

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{3}{4} (1 \cdot 4)$

Step 3.1.1.3.2.6

Cancel the common factor of $4$ .

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Step 3.1.1.3.2.6.1

Factor $4$ out of $1 \cdot 4$ .

$d = \frac{3}{4} (4 \cdot 1)$

Step 3.1.1.3.2.6.2

Cancel the common factor.

$d = \frac{3}{4} (4 \cdot 1)$

Step 3.1.1.3.2.6.3

Rewrite the expression.

$d = 3$

Step 3.1.1.4

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

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Step 3.1.1.4.1

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

$e = - \frac{25}{8} - \frac{{(- \frac{3}{4})}^{2}}{4 (- \frac{1}{8})}$

Step 3.1.1.4.2

Simplify the right side.

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Step 3.1.1.4.2.1

Simplify each term.

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Step 3.1.1.4.2.1.1

Simplify the numerator.

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Step 3.1.1.4.2.1.1.1

Apply the product rule to $- \frac{3}{4}$ .

$e = - \frac{25}{8} - \frac{{(- 1)}^{2} {(\frac{3}{4})}^{2}}{4 (- \frac{1}{8})}$

Step 3.1.1.4.2.1.1.2

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

$e = - \frac{25}{8} - \frac{1 {(\frac{3}{4})}^{2}}{4 (- \frac{1}{8})}$

Step 3.1.1.4.2.1.1.3

Apply the product rule to $\frac{3}{4}$ .

$e = - \frac{25}{8} - \frac{1 \frac{3^{2}}{4^{2}}}{4 (- \frac{1}{8})}$

Step 3.1.1.4.2.1.1.4

Raise $3$ to the power of $2$ .

$e = - \frac{25}{8} - \frac{1 \frac{9}{4^{2}}}{4 (- \frac{1}{8})}$

Step 3.1.1.4.2.1.1.5

Raise $4$ to the power of $2$ .

$e = - \frac{25}{8} - \frac{1 (\frac{9}{16})}{4 (- \frac{1}{8})}$

Step 3.1.1.4.2.1.1.6

Multiply $\frac{9}{16}$ by $1$ .

$e = - \frac{25}{8} - \frac{\frac{9}{16}}{4 (- \frac{1}{8})}$

Step 3.1.1.4.2.1.2

Simplify the denominator.

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Step 3.1.1.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = - \frac{25}{8} - \frac{\frac{9}{16}}{- 4 (\frac{1}{8})}$

Step 3.1.1.4.2.1.2.2

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

$e = - \frac{25}{8} - \frac{\frac{9}{16}}{\frac{- 4}{8}}$

Step 3.1.1.4.2.1.3

Reduce the expression by cancelling the common factors.

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Step 3.1.1.4.2.1.3.1

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

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Step 3.1.1.4.2.1.3.1.1

Factor $4$ out of $- 4$ .

$e = - \frac{25}{8} - \frac{\frac{9}{16}}{\frac{4 (- 1)}{8}}$

Step 3.1.1.4.2.1.3.1.2

Cancel the common factors.

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Step 3.1.1.4.2.1.3.1.2.1

Factor $4$ out of $8$ .

$e = - \frac{25}{8} - \frac{\frac{9}{16}}{\frac{4 \cdot - 1}{4 \cdot 2}}$

Step 3.1.1.4.2.1.3.1.2.2

Cancel the common factor.

$e = - \frac{25}{8} - \frac{\frac{9}{16}}{\frac{4 \cdot - 1}{4 \cdot 2}}$

Step 3.1.1.4.2.1.3.1.2.3

Rewrite the expression.

$e = - \frac{25}{8} - \frac{\frac{9}{16}}{\frac{- 1}{2}}$

Step 3.1.1.4.2.1.3.2

Move the negative in front of the fraction.

$e = - \frac{25}{8} - \frac{\frac{9}{16}}{- \frac{1}{2}}$

Step 3.1.1.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{25}{8} - (\frac{9}{16} (- 1 \cdot 2))$

Step 3.1.1.4.2.1.5

Cancel the common factor of $2$ .

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Step 3.1.1.4.2.1.5.1

Factor $2$ out of $16$ .

$e = - \frac{25}{8} - (\frac{9}{2 (8)} (- 1 \cdot 2))$

Step 3.1.1.4.2.1.5.2

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

$e = - \frac{25}{8} - (\frac{9}{2 (8)} (2 \cdot - 1))$

Step 3.1.1.4.2.1.5.3

Cancel the common factor.

$e = - \frac{25}{8} - (\frac{9}{2 \cdot 8} (2 \cdot - 1))$

Step 3.1.1.4.2.1.5.4

Rewrite the expression.

$e = - \frac{25}{8} - (\frac{9}{8} \cdot - 1)$

Step 3.1.1.4.2.1.6

Combine $\frac{9}{8}$ and $- 1$ .

$e = - \frac{25}{8} - \frac{9 \cdot - 1}{8}$

Step 3.1.1.4.2.1.7

Multiply $9$ by $- 1$ .

$e = - \frac{25}{8} - \frac{- 9}{8}$

Step 3.1.1.4.2.1.8

Move the negative in front of the fraction.

$e = - \frac{25}{8} - - \frac{9}{8}$

Step 3.1.1.4.2.1.9

Multiply $- - \frac{9}{8}$ .

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Step 3.1.1.4.2.1.9.1

Multiply $- 1$ by $- 1$ .

$e = - \frac{25}{8} + 1 (\frac{9}{8})$

Step 3.1.1.4.2.1.9.2

Multiply $\frac{9}{8}$ by $1$ .

$e = - \frac{25}{8} + \frac{9}{8}$

Step 3.1.1.4.2.2

Combine the numerators over the common denominator.

$e = \frac{- 25 + 9}{8}$

Step 3.1.1.4.2.3

Add $- 25$ and $9$ .

$e = \frac{- 16}{8}$

Step 3.1.1.4.2.4

Divide $- 16$ by $8$ .

$e = - 2$

Step 3.1.1.5

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

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

Step 3.1.2

Set $x$ equal to the new right side.

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

Step 3.2

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

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

$h = - 2$

$k = - 3$

Step 3.3

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

Opens Left

Step 3.4

Find the vertex $(h, k)$ .

$(- 2, - 3)$

Step 3.5

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

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Step 3.5.1

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

$\frac{1}{4 a}$

Step 3.5.2

Substitute the value of $a$ into the formula.

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

Step 3.5.3

Simplify.

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Step 3.5.3.1

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

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Step 3.5.3.1.1

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

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

Step 3.5.3.1.2

Move the negative in front of the fraction.

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

Step 3.5.3.2

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

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

Step 3.5.3.3

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

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Step 3.5.3.3.1

Factor $4$ out of $4$ .

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

Step 3.5.3.3.2

Cancel the common factors.

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Step 3.5.3.3.2.1

Factor $4$ out of $8$ .

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

Step 3.5.3.3.2.2

Cancel the common factor.

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

Step 3.5.3.3.2.3

Rewrite the expression.

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

Step 3.5.3.4

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 2)$

Step 3.5.3.5

Multiply $- (1 \cdot 2)$ .

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Step 3.5.3.5.1

Multiply $2$ by $1$ .

$- 1 \cdot 2$

Step 3.5.3.5.2

Multiply $- 1$ by $2$ .

$- 2$

Step 3.6

Find the focus.

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Step 3.6.1

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 3.6.2

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

$(- 4, - 3)$

Step 3.7

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

$y = - 3$

Step 3.8

Find the directrix.

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Step 3.8.1

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 3.8.2

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

$x = 0$

Step 3.9

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

Direction: Opens Left

Vertex: $(- 2, - 3)$

Focus: $(- 4, - 3)$

Axis of Symmetry: $y = - 3$

Directrix: $x = 0$

Direction: Opens Left

Vertex: $(- 2, - 3)$

Focus: $(- 4, - 3)$

Axis of Symmetry: $y = - 3$

Directrix: $x = 0$

Step 4

Select a few $x$ values, and plug them into the equation to find the corresponding $y$ values. The $x$ values should be selected around the vertex.

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Step 4.1

Substitute the $x$ value $- 4$ into $f (x) = - 3 + 2 \sqrt{2 (- x - 2)}$ . In this case, the point is $(- 4, 1)$ .

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Step 4.1.1

Replace the variable $x$ with $- 4$ in the expression.

$f (- 4) = - 3 + 2 \sqrt{2 (- (- 4) - 2)}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Multiply $- 1$ by $- 4$ .

$f (- 4) = - 3 + 2 \sqrt{2 (4 - 2)}$

Step 4.1.2.1.2

Subtract $2$ from $4$ .

$f (- 4) = - 3 + 2 \sqrt{2 \cdot 2}$

Step 4.1.2.1.3

Multiply $2$ by $2$ .

$f (- 4) = - 3 + 2 \sqrt{4}$

Step 4.1.2.1.4

Rewrite $4$ as $2^{2}$ .

$f (- 4) = - 3 + 2 \sqrt{2^{2}}$

Step 4.1.2.1.5

Pull terms out from under the radical, assuming positive real numbers.

$f (- 4) = - 3 + 2 \cdot 2$

Step 4.1.2.1.6

Multiply $2$ by $2$ .

$f (- 4) = - 3 + 4$

Step 4.1.2.2

Add $- 3$ and $4$ .

$f (- 4) = 1$

Step 4.1.2.3

The final answer is $1$ .

$y = 1$

Step 4.1.3

Convert $1$ to decimal.

$= 1$

Step 4.2

Substitute the $x$ value $- 4$ into $f (x) = - 3 - 2 \sqrt{2 (- x - 2)}$ . In this case, the point is $(- 4, - 7)$ .

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Step 4.2.1

Replace the variable $x$ with $- 4$ in the expression.

$f (- 4) = - 3 - 2 \sqrt{2 (- (- 4) - 2)}$

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Multiply $- 1$ by $- 4$ .

$f (- 4) = - 3 - 2 \sqrt{2 (4 - 2)}$

Step 4.2.2.1.2

Subtract $2$ from $4$ .

$f (- 4) = - 3 - 2 \sqrt{2 \cdot 2}$

Step 4.2.2.1.3

Multiply $2$ by $2$ .

$f (- 4) = - 3 - 2 \sqrt{4}$

Step 4.2.2.1.4

Rewrite $4$ as $2^{2}$ .

$f (- 4) = - 3 - 2 \sqrt{2^{2}}$

Step 4.2.2.1.5

Pull terms out from under the radical, assuming positive real numbers.

$f (- 4) = - 3 - 2 \cdot 2$

Step 4.2.2.1.6

Multiply $- 2$ by $2$ .

$f (- 4) = - 3 - 4$

Step 4.2.2.2

Subtract $4$ from $- 3$ .

$f (- 4) = - 7$

Step 4.2.2.3

The final answer is $- 7$ .

$y = - 7$

Step 4.2.3

Convert $- 7$ to decimal.

$= - 7$

Step 4.3

Substitute the $x$ value $- 3$ into $f (x) = - 3 + 2 \sqrt{2 (- x - 2)}$ . In this case, the point is $(- 3, - 0.17157287)$ .

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Step 4.3.1

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = - 3 + 2 \sqrt{2 (- (- 3) - 2)}$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Multiply $- 1$ by $- 3$ .

$f (- 3) = - 3 + 2 \sqrt{2 (3 - 2)}$

Step 4.3.2.1.2

Subtract $2$ from $3$ .

$f (- 3) = - 3 + 2 \sqrt{2 \cdot 1}$

Step 4.3.2.1.3

Multiply $2$ by $1$ .

$f (- 3) = - 3 + 2 \sqrt{2}$

Step 4.3.2.2

The final answer is $- 3 + 2 \sqrt{2}$ .

$y = - 3 + 2 \sqrt{2}$

Step 4.3.3

Convert $- 3 + 2 \sqrt{2}$ to decimal.

$= - 0.17157287$

Step 4.4

Substitute the $x$ value $- 3$ into $f (x) = - 3 - 2 \sqrt{2 (- x - 2)}$ . In this case, the point is $(- 3, - 5.82842712)$ .

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Step 4.4.1

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = - 3 - 2 \sqrt{2 (- (- 3) - 2)}$

Step 4.4.2

Simplify the result.

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Step 4.4.2.1

Simplify each term.

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Step 4.4.2.1.1

Multiply $- 1$ by $- 3$ .

$f (- 3) = - 3 - 2 \sqrt{2 (3 - 2)}$

Step 4.4.2.1.2

Subtract $2$ from $3$ .

$f (- 3) = - 3 - 2 \sqrt{2 \cdot 1}$

Step 4.4.2.1.3

Multiply $2$ by $1$ .

$f (- 3) = - 3 - 2 \sqrt{2}$

Step 4.4.2.2

The final answer is $- 3 - 2 \sqrt{2}$ .

$y = - 3 - 2 \sqrt{2}$

Step 4.4.3

Convert $- 3 - 2 \sqrt{2}$ to decimal.

$= - 5.82842712$

Step 4.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 4 & 1 \\ - 4 & - 7 \\ - 3 & - 0.17 \\ - 3 & - 5.83 \\ - 2 & - 3 \end{array}$

Step 5

Graph the parabola using its properties and the selected points.

Direction: Opens Left

Vertex: $(- 2, - 3)$

Focus: $(- 4, - 3)$

Axis of Symmetry: $y = - 3$

Directrix: $x = 0$

$\begin{array}{c} x & y \\ - 4 & 1 \\ - 4 & - 7 \\ - 3 & - 0.17 \\ - 3 & - 5.83 \\ - 2 & - 3 \end{array}$

Step 6