Precalculus Examples

${(x + 5)}^{2} = 24 (y + 1)$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

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Step 1.1.1

Rewrite the equation as $24 (y + 1) = {(x + 5)}^{2}$ .

$24 (y + 1) = {(x + 5)}^{2}$

Step 1.1.2

Divide each term in $24 (y + 1) = {(x + 5)}^{2}$ by $24$ and simplify.

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Step 1.1.2.1

Divide each term in $24 (y + 1) = {(x + 5)}^{2}$ by $24$ .

$\frac{24 (y + 1)}{24} = \frac{{(x + 5)}^{2}}{24}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $24$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{24 (y + 1)}{24} = \frac{{(x + 5)}^{2}}{24}$

Step 1.1.2.2.1.2

Divide $y + 1$ by $1$ .

$y + 1 = \frac{{(x + 5)}^{2}}{24}$

Step 1.1.3

Subtract $1$ from both sides of the equation.

$y = \frac{{(x + 5)}^{2}}{24} - 1$

Step 1.1.4

Reorder terms.

$y = \frac{1}{24} \cdot {(x + 5)}^{2} - 1$

Step 1.2

Complete the square for $\frac{1}{24} \cdot {(x + 5)}^{2} - 1$ .

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Step 1.2.1

Simplify the expression.

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

$\frac{1}{24} \cdot ((x + 5) (x + 5)) - 1$

Step 1.2.1.1.2

Expand $(x + 5) (x + 5)$ using the FOIL Method.

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Step 1.2.1.1.2.1

Apply the distributive property.

$\frac{1}{24} \cdot (x (x + 5) + 5 (x + 5)) - 1$

Step 1.2.1.1.2.2

Apply the distributive property.

$\frac{1}{24} \cdot (x \cdot x + x \cdot 5 + 5 (x + 5)) - 1$

Step 1.2.1.1.2.3

Apply the distributive property.

$\frac{1}{24} \cdot (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5) - 1$

Step 1.2.1.1.3

Simplify and combine like terms.

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Step 1.2.1.1.3.1

Simplify each term.

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Step 1.2.1.1.3.1.1

Multiply $x$ by $x$ .

$\frac{1}{24} \cdot (x^{2} + x \cdot 5 + 5 x + 5 \cdot 5) - 1$

Step 1.2.1.1.3.1.2

Move $5$ to the left of $x$ .

$\frac{1}{24} \cdot (x^{2} + 5 \cdot x + 5 x + 5 \cdot 5) - 1$

Step 1.2.1.1.3.1.3

Multiply $5$ by $5$ .

$\frac{1}{24} \cdot (x^{2} + 5 x + 5 x + 25) - 1$

Step 1.2.1.1.3.2

Add $5 x$ and $5 x$ .

$\frac{1}{24} \cdot (x^{2} + 10 x + 25) - 1$

Step 1.2.1.1.4

Apply the distributive property.

$\frac{1}{24} x^{2} + \frac{1}{24} (10 x) + \frac{1}{24} \cdot 25 - 1$

Step 1.2.1.1.5

Simplify.

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Step 1.2.1.1.5.1

Combine $\frac{1}{24}$ and $x^{2}$ .

$\frac{x^{2}}{24} + \frac{1}{24} (10 x) + \frac{1}{24} \cdot 25 - 1$

Step 1.2.1.1.5.2

Cancel the common factor of $2$ .

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Step 1.2.1.1.5.2.1

Factor $2$ out of $24$ .

$\frac{x^{2}}{24} + \frac{1}{2 (12)} (10 x) + \frac{1}{24} \cdot 25 - 1$

Step 1.2.1.1.5.2.2

Factor $2$ out of $10 x$ .

$\frac{x^{2}}{24} + \frac{1}{2 (12)} (2 (5 x)) + \frac{1}{24} \cdot 25 - 1$

Step 1.2.1.1.5.2.3

Cancel the common factor.

$\frac{x^{2}}{24} + \frac{1}{2 \cdot 12} (2 (5 x)) + \frac{1}{24} \cdot 25 - 1$

Step 1.2.1.1.5.2.4

Rewrite the expression.

$\frac{x^{2}}{24} + \frac{1}{12} (5 x) + \frac{1}{24} \cdot 25 - 1$

Step 1.2.1.1.5.3

Combine $5$ and $\frac{1}{12}$ .

$\frac{x^{2}}{24} + \frac{5}{12} x + \frac{1}{24} \cdot 25 - 1$

Step 1.2.1.1.5.4

Combine $\frac{5}{12}$ and $x$ .

$\frac{x^{2}}{24} + \frac{5 x}{12} + \frac{1}{24} \cdot 25 - 1$

Step 1.2.1.1.5.5

Combine $\frac{1}{24}$ and $25$ .

$\frac{x^{2}}{24} + \frac{5 x}{12} + \frac{25}{24} - 1$

Step 1.2.1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{24}{24}$ .

$\frac{x^{2}}{24} + \frac{5 x}{12} + \frac{25}{24} - 1 \cdot \frac{24}{24}$

Step 1.2.1.3

Combine $- 1$ and $\frac{24}{24}$ .

$\frac{x^{2}}{24} + \frac{5 x}{12} + \frac{25}{24} + \frac{- 1 \cdot 24}{24}$

Step 1.2.1.4

Combine the numerators over the common denominator.

$\frac{x^{2}}{24} + \frac{5 x}{12} + \frac{25 - 1 \cdot 24}{24}$

Step 1.2.1.5

Simplify the numerator.

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Step 1.2.1.5.1

Multiply $- 1$ by $24$ .

$\frac{x^{2}}{24} + \frac{5 x}{12} + \frac{25 - 24}{24}$

Step 1.2.1.5.2

Subtract $24$ from $25$ .

$\frac{x^{2}}{24} + \frac{5 x}{12} + \frac{1}{24}$

Step 1.2.2

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

$a = \frac{1}{24}$

$b = \frac{5}{12}$

$c = \frac{1}{24}$

Step 1.2.3

Consider the vertex form of a parabola.

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

Step 1.2.4

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

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Step 1.2.4.1

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

$d = \frac{\frac{5}{12}}{2 (\frac{1}{24})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{5}{12} \cdot \frac{1}{2 (\frac{1}{24})}$

Step 1.2.4.2.2

Combine $2$ and $\frac{1}{24}$ .

$d = \frac{5}{12} \cdot \frac{1}{\frac{2}{24}}$

Step 1.2.4.2.3

Cancel the common factor of $2$ and $24$ .

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Step 1.2.4.2.3.1

Factor $2$ out of $2$ .

$d = \frac{5}{12} \cdot \frac{1}{\frac{2 (1)}{24}}$

Step 1.2.4.2.3.2

Cancel the common factors.

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Step 1.2.4.2.3.2.1

Factor $2$ out of $24$ .

$d = \frac{5}{12} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 12}}$

Step 1.2.4.2.3.2.2

Cancel the common factor.

$d = \frac{5}{12} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 12}}$

Step 1.2.4.2.3.2.3

Rewrite the expression.

$d = \frac{5}{12} \cdot \frac{1}{\frac{1}{12}}$

Step 1.2.4.2.4

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{5}{12} (1 \cdot 12)$

Step 1.2.4.2.5

Cancel the common factor of $12$ .

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Step 1.2.4.2.5.1

Factor $12$ out of $1 \cdot 12$ .

$d = \frac{5}{12} (12 \cdot 1)$

Step 1.2.4.2.5.2

Cancel the common factor.

$d = \frac{5}{12} (12 \cdot 1)$

Step 1.2.4.2.5.3

Rewrite the expression.

$d = 5$

Step 1.2.5

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

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Step 1.2.5.1

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

$e = \frac{1}{24} - \frac{{(\frac{5}{12})}^{2}}{4 (\frac{1}{24})}$

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Simplify each term.

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Step 1.2.5.2.1.1

Simplify the numerator.

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Step 1.2.5.2.1.1.1

Apply the product rule to $\frac{5}{12}$ .

$e = \frac{1}{24} - \frac{\frac{5^{2}}{12^{2}}}{4 (\frac{1}{24})}$

Step 1.2.5.2.1.1.2

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

$e = \frac{1}{24} - \frac{\frac{25}{12^{2}}}{4 (\frac{1}{24})}$

Step 1.2.5.2.1.1.3

Raise $12$ to the power of $2$ .

$e = \frac{1}{24} - \frac{\frac{25}{144}}{4 (\frac{1}{24})}$

Step 1.2.5.2.1.2

Combine $4$ and $\frac{1}{24}$ .

$e = \frac{1}{24} - \frac{\frac{25}{144}}{\frac{4}{24}}$

Step 1.2.5.2.1.3

Cancel the common factor of $4$ and $24$ .

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Step 1.2.5.2.1.3.1

Factor $4$ out of $4$ .

$e = \frac{1}{24} - \frac{\frac{25}{144}}{\frac{4 (1)}{24}}$

Step 1.2.5.2.1.3.2

Cancel the common factors.

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Step 1.2.5.2.1.3.2.1

Factor $4$ out of $24$ .

$e = \frac{1}{24} - \frac{\frac{25}{144}}{\frac{4 \cdot 1}{4 \cdot 6}}$

Step 1.2.5.2.1.3.2.2

Cancel the common factor.

$e = \frac{1}{24} - \frac{\frac{25}{144}}{\frac{4 \cdot 1}{4 \cdot 6}}$

Step 1.2.5.2.1.3.2.3

Rewrite the expression.

$e = \frac{1}{24} - \frac{\frac{25}{144}}{\frac{1}{6}}$

Step 1.2.5.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{1}{24} - (\frac{25}{144} \cdot 6)$

Step 1.2.5.2.1.5

Cancel the common factor of $6$ .

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Step 1.2.5.2.1.5.1

Factor $6$ out of $144$ .

$e = \frac{1}{24} - (\frac{25}{6 (24)} \cdot 6)$

Step 1.2.5.2.1.5.2

Cancel the common factor.

$e = \frac{1}{24} - (\frac{25}{6 \cdot 24} \cdot 6)$

Step 1.2.5.2.1.5.3

Rewrite the expression.

$e = \frac{1}{24} - \frac{25}{24}$

Step 1.2.5.2.2

Combine the numerators over the common denominator.

$e = \frac{1 - 25}{24}$

Step 1.2.5.2.3

Subtract $25$ from $1$ .

$e = \frac{- 24}{24}$

Step 1.2.5.2.4

Divide $- 24$ by $24$ .

$e = - 1$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{24} {(x + 5)}^{2} - 1$ .

$\frac{1}{24} {(x + 5)}^{2} - 1$

Step 1.3

Set $y$ equal to the new right side.

$y = \frac{1}{24} \cdot {(x + 5)}^{2} - 1$

Step 2

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

$a = \frac{1}{24}$

$h = - 5$

$k = - 1$

Step 3

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

Opens Up

Step 4

Find the vertex $(h, k)$ .

$(- 5, - 1)$

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.

$\frac{1}{4 \cdot \frac{1}{24}}$

Step 5.3

Simplify.

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Step 5.3.1

Combine $4$ and $\frac{1}{24}$ .

$\frac{1}{\frac{4}{24}}$

Step 5.3.2

Cancel the common factor of $4$ and $24$ .

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Step 5.3.2.1

Factor $4$ out of $4$ .

$\frac{1}{\frac{4 (1)}{24}}$

Step 5.3.2.2

Cancel the common factors.

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Step 5.3.2.2.1

Factor $4$ out of $24$ .

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 6}}$

Step 5.3.2.2.2

Cancel the common factor.

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 6}}$

Step 5.3.2.2.3

Rewrite the expression.

$\frac{1}{\frac{1}{6}}$

Step 5.3.3

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 6$

Step 5.3.4

Multiply $6$ by $1$ .

$6$

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 y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Step 6.2

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

$(- 5, 5)$

Step 7

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

$x = - 5$

Step 8

Find the directrix.

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Step 8.1

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Step 8.2

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

$y = - 7$

Step 9

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

Direction: Opens Up

Vertex: $(- 5, - 1)$

Focus: $(- 5, 5)$

Axis of Symmetry: $x = - 5$

Directrix: $y = - 7$

Step 10