Precalculus Examples

${(x - 5)}^{2} = 16 (y + 1)$

Step 1

Solve for $y$ .

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Step 1.1

Rewrite the equation as $16 (y + 1) = {(x - 5)}^{2}$ .

$16 (y + 1) = {(x - 5)}^{2}$

Step 1.2

Divide each term in $16 (y + 1) = {(x - 5)}^{2}$ by $16$ and simplify.

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Step 1.2.1

Divide each term in $16 (y + 1) = {(x - 5)}^{2}$ by $16$ .

$\frac{16 (y + 1)}{16} = \frac{{(x - 5)}^{2}}{16}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $16$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{16 (y + 1)}{16} = \frac{{(x - 5)}^{2}}{16}$

Step 1.2.2.1.2

Divide $y + 1$ by $1$ .

$y + 1 = \frac{{(x - 5)}^{2}}{16}$

Step 1.3

Subtract $1$ from both sides of the equation.

$y = \frac{{(x - 5)}^{2}}{16} - 1$

Step 2

Find the properties of the given parabola.

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Step 2.1

Rewrite the equation in vertex form.

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Step 2.1.1

Reorder terms.

$y = \frac{1}{16} \cdot {(x - 5)}^{2} - 1$

Step 2.1.2

Complete the square for $\frac{1}{16} \cdot {(x - 5)}^{2} - 1$ .

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Step 2.1.2.1

Simplify the expression.

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Step 2.1.2.1.1

Simplify each term.

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Step 2.1.2.1.1.1

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

$\frac{1}{16} \cdot ((x - 5) (x - 5)) - 1$

Step 2.1.2.1.1.2

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

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Step 2.1.2.1.1.2.1

Apply the distributive property.

$\frac{1}{16} \cdot (x (x - 5) - 5 (x - 5)) - 1$

Step 2.1.2.1.1.2.2

Apply the distributive property.

$\frac{1}{16} \cdot (x \cdot x + x \cdot - 5 - 5 (x - 5)) - 1$

Step 2.1.2.1.1.2.3

Apply the distributive property.

$\frac{1}{16} \cdot (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5) - 1$

Step 2.1.2.1.1.3

Simplify and combine like terms.

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Step 2.1.2.1.1.3.1

Simplify each term.

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Step 2.1.2.1.1.3.1.1

Multiply $x$ by $x$ .

$\frac{1}{16} \cdot (x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5) - 1$

Step 2.1.2.1.1.3.1.2

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

$\frac{1}{16} \cdot (x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5) - 1$

Step 2.1.2.1.1.3.1.3

Multiply $- 5$ by $- 5$ .

$\frac{1}{16} \cdot (x^{2} - 5 x - 5 x + 25) - 1$

Step 2.1.2.1.1.3.2

Subtract $5 x$ from $- 5 x$ .

$\frac{1}{16} \cdot (x^{2} - 10 x + 25) - 1$

Step 2.1.2.1.1.4

Apply the distributive property.

$\frac{1}{16} x^{2} + \frac{1}{16} (- 10 x) + \frac{1}{16} \cdot 25 - 1$

Step 2.1.2.1.1.5

Simplify.

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Step 2.1.2.1.1.5.1

Combine $\frac{1}{16}$ and $x^{2}$ .

$\frac{x^{2}}{16} + \frac{1}{16} (- 10 x) + \frac{1}{16} \cdot 25 - 1$

Step 2.1.2.1.1.5.2

Cancel the common factor of $2$ .

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Step 2.1.2.1.1.5.2.1

Factor $2$ out of $16$ .

$\frac{x^{2}}{16} + \frac{1}{2 (8)} (- 10 x) + \frac{1}{16} \cdot 25 - 1$

Step 2.1.2.1.1.5.2.2

Factor $2$ out of $- 10 x$ .

$\frac{x^{2}}{16} + \frac{1}{2 (8)} (2 (- 5 x)) + \frac{1}{16} \cdot 25 - 1$

Step 2.1.2.1.1.5.2.3

Cancel the common factor.

$\frac{x^{2}}{16} + \frac{1}{2 \cdot 8} (2 (- 5 x)) + \frac{1}{16} \cdot 25 - 1$

Step 2.1.2.1.1.5.2.4

Rewrite the expression.

$\frac{x^{2}}{16} + \frac{1}{8} (- 5 x) + \frac{1}{16} \cdot 25 - 1$

Step 2.1.2.1.1.5.3

Combine $- 5$ and $\frac{1}{8}$ .

$\frac{x^{2}}{16} + \frac{- 5}{8} x + \frac{1}{16} \cdot 25 - 1$

Step 2.1.2.1.1.5.4

Combine $\frac{- 5}{8}$ and $x$ .

$\frac{x^{2}}{16} + \frac{- 5 x}{8} + \frac{1}{16} \cdot 25 - 1$

Step 2.1.2.1.1.5.5

Combine $\frac{1}{16}$ and $25$ .

$\frac{x^{2}}{16} + \frac{- 5 x}{8} + \frac{25}{16} - 1$

Step 2.1.2.1.1.6

Move the negative in front of the fraction.

$\frac{x^{2}}{16} - \frac{5 x}{8} + \frac{25}{16} - 1$

Step 2.1.2.1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\frac{x^{2}}{16} - \frac{5 x}{8} + \frac{25}{16} - 1 \cdot \frac{16}{16}$

Step 2.1.2.1.3

Combine $- 1$ and $\frac{16}{16}$ .

$\frac{x^{2}}{16} - \frac{5 x}{8} + \frac{25}{16} + \frac{- 1 \cdot 16}{16}$

Step 2.1.2.1.4

Combine the numerators over the common denominator.

$\frac{x^{2}}{16} - \frac{5 x}{8} + \frac{25 - 1 \cdot 16}{16}$

Step 2.1.2.1.5

Simplify the numerator.

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Step 2.1.2.1.5.1

Multiply $- 1$ by $16$ .

$\frac{x^{2}}{16} - \frac{5 x}{8} + \frac{25 - 16}{16}$

Step 2.1.2.1.5.2

Subtract $16$ from $25$ .

$\frac{x^{2}}{16} - \frac{5 x}{8} + \frac{9}{16}$

Step 2.1.2.2

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

$a = \frac{1}{16}$

$b = - \frac{5}{8}$

$c = \frac{9}{16}$

Step 2.1.2.3

Consider the vertex form of a parabola.

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

Step 2.1.2.4

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

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Step 2.1.2.4.1

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

$d = \frac{- \frac{5}{8}}{2 (\frac{1}{16})}$

Step 2.1.2.4.2

Simplify the right side.

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Step 2.1.2.4.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{5}{8} \cdot \frac{1}{2 (\frac{1}{16})}$

Step 2.1.2.4.2.2

Combine $2$ and $\frac{1}{16}$ .

$d = - \frac{5}{8} \cdot \frac{1}{\frac{2}{16}}$

Step 2.1.2.4.2.3

Cancel the common factor of $2$ and $16$ .

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Step 2.1.2.4.2.3.1

Factor $2$ out of $2$ .

$d = - \frac{5}{8} \cdot \frac{1}{\frac{2 (1)}{16}}$

Step 2.1.2.4.2.3.2

Cancel the common factors.

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Step 2.1.2.4.2.3.2.1

Factor $2$ out of $16$ .

$d = - \frac{5}{8} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 8}}$

Step 2.1.2.4.2.3.2.2

Cancel the common factor.

$d = - \frac{5}{8} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 8}}$

Step 2.1.2.4.2.3.2.3

Rewrite the expression.

$d = - \frac{5}{8} \cdot \frac{1}{\frac{1}{8}}$

Step 2.1.2.4.2.4

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{5}{8} (1 \cdot 8)$

Step 2.1.2.4.2.5

Cancel the common factor of $8$ .

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Step 2.1.2.4.2.5.1

Move the leading negative in $- \frac{5}{8}$ into the numerator.

$d = \frac{- 5}{8} (1 \cdot 8)$

Step 2.1.2.4.2.5.2

Factor $8$ out of $1 \cdot 8$ .

$d = \frac{- 5}{8} (8 \cdot 1)$

Step 2.1.2.4.2.5.3

Cancel the common factor.

$d = \frac{- 5}{8} (8 \cdot 1)$

Step 2.1.2.4.2.5.4

Rewrite the expression.

$d = - 5$

Step 2.1.2.5

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

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Step 2.1.2.5.1

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

$e = \frac{9}{16} - \frac{{(- \frac{5}{8})}^{2}}{4 (\frac{1}{16})}$

Step 2.1.2.5.2

Simplify the right side.

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Step 2.1.2.5.2.1

Simplify each term.

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Step 2.1.2.5.2.1.1

Simplify the numerator.

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Step 2.1.2.5.2.1.1.1

Apply the product rule to $- \frac{5}{8}$ .

$e = \frac{9}{16} - \frac{{(- 1)}^{2} {(\frac{5}{8})}^{2}}{4 (\frac{1}{16})}$

Step 2.1.2.5.2.1.1.2

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

$e = \frac{9}{16} - \frac{1 {(\frac{5}{8})}^{2}}{4 (\frac{1}{16})}$

Step 2.1.2.5.2.1.1.3

Apply the product rule to $\frac{5}{8}$ .

$e = \frac{9}{16} - \frac{1 \frac{5^{2}}{8^{2}}}{4 (\frac{1}{16})}$

Step 2.1.2.5.2.1.1.4

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

$e = \frac{9}{16} - \frac{1 \frac{25}{8^{2}}}{4 (\frac{1}{16})}$

Step 2.1.2.5.2.1.1.5

Raise $8$ to the power of $2$ .

$e = \frac{9}{16} - \frac{1 (\frac{25}{64})}{4 (\frac{1}{16})}$

Step 2.1.2.5.2.1.1.6

Multiply $\frac{25}{64}$ by $1$ .

$e = \frac{9}{16} - \frac{\frac{25}{64}}{4 (\frac{1}{16})}$

Step 2.1.2.5.2.1.2

Combine $4$ and $\frac{1}{16}$ .

$e = \frac{9}{16} - \frac{\frac{25}{64}}{\frac{4}{16}}$

Step 2.1.2.5.2.1.3

Cancel the common factor of $4$ and $16$ .

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Step 2.1.2.5.2.1.3.1

Factor $4$ out of $4$ .

$e = \frac{9}{16} - \frac{\frac{25}{64}}{\frac{4 (1)}{16}}$

Step 2.1.2.5.2.1.3.2

Cancel the common factors.

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Step 2.1.2.5.2.1.3.2.1

Factor $4$ out of $16$ .

$e = \frac{9}{16} - \frac{\frac{25}{64}}{\frac{4 \cdot 1}{4 \cdot 4}}$

Step 2.1.2.5.2.1.3.2.2

Cancel the common factor.

$e = \frac{9}{16} - \frac{\frac{25}{64}}{\frac{4 \cdot 1}{4 \cdot 4}}$

Step 2.1.2.5.2.1.3.2.3

Rewrite the expression.

$e = \frac{9}{16} - \frac{\frac{25}{64}}{\frac{1}{4}}$

Step 2.1.2.5.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{9}{16} - (\frac{25}{64} \cdot 4)$

Step 2.1.2.5.2.1.5

Cancel the common factor of $4$ .

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Step 2.1.2.5.2.1.5.1

Factor $4$ out of $64$ .

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

Step 2.1.2.5.2.1.5.2

Cancel the common factor.

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

Step 2.1.2.5.2.1.5.3

Rewrite the expression.

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

Step 2.1.2.5.2.2

Combine the numerators over the common denominator.

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

Step 2.1.2.5.2.3

Subtract $25$ from $9$ .

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

Step 2.1.2.5.2.4

Divide $- 16$ by $16$ .

$e = - 1$

Step 2.1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{16} {(x - 5)}^{2} - 1$ .

$\frac{1}{16} {(x - 5)}^{2} - 1$

Step 2.1.3

Set $y$ equal to the new right side.

$y = \frac{1}{16} \cdot {(x - 5)}^{2} - 1$

Step 2.2

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

$a = \frac{1}{16}$

$h = 5$

$k = - 1$

Step 2.3

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

Opens Up

Step 2.4

Find the vertex $(h, k)$ .

$(5, - 1)$

Step 2.5

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

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Step 2.5.1

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

$\frac{1}{4 a}$

Step 2.5.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot \frac{1}{16}}$

Step 2.5.3

Simplify.

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Step 2.5.3.1

Combine $4$ and $\frac{1}{16}$ .

$\frac{1}{\frac{4}{16}}$

Step 2.5.3.2

Cancel the common factor of $4$ and $16$ .

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Step 2.5.3.2.1

Factor $4$ out of $4$ .

$\frac{1}{\frac{4 (1)}{16}}$

Step 2.5.3.2.2

Cancel the common factors.

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Step 2.5.3.2.2.1

Factor $4$ out of $16$ .

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

Step 2.5.3.2.2.2

Cancel the common factor.

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

Step 2.5.3.2.2.3

Rewrite the expression.

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

Step 2.5.3.3

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 4$

Step 2.5.3.4

Multiply $4$ by $1$ .

$4$

Step 2.6

Find the focus.

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Step 2.6.1

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 2.6.2

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

$(5, 3)$

Step 2.7

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

$x = 5$

Step 2.8

Find the directrix.

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Step 2.8.1

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 2.8.2

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

$y = - 5$

Step 2.9

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

Direction: Opens Up

Vertex: $(5, - 1)$

Focus: $(5, 3)$

Axis of Symmetry: $x = 5$

Directrix: $y = - 5$

Direction: Opens Up

Vertex: $(5, - 1)$

Focus: $(5, 3)$

Axis of Symmetry: $x = 5$

Directrix: $y = - 5$

Step 3

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 3.1

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

$f (4) = \frac{((4) - 1) ((4) - 9)}{16}$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify the numerator.

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Step 3.2.1.1

Subtract $1$ from $4$ .

$f (4) = \frac{3 (4 - 9)}{16}$

Step 3.2.1.2

Subtract $9$ from $4$ .

$f (4) = \frac{3 \cdot - 5}{16}$

Step 3.2.2

Simplify the expression.

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Step 3.2.2.1

Multiply $3$ by $- 5$ .

$f (4) = \frac{- 15}{16}$

Step 3.2.2.2

Move the negative in front of the fraction.

$f (4) = - \frac{15}{16}$

Step 3.2.3

The final answer is $- \frac{15}{16}$ .

$- \frac{15}{16}$

Step 3.3

The $y$ value at $x = 4$ is $- \frac{15}{16}$ .

$y = - \frac{15}{16}$

Step 3.4

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

$f (3) = \frac{((3) - 1) ((3) - 9)}{16}$

Step 3.5

Simplify the result.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Subtract $1$ from $3$ .

$f (3) = \frac{2 (3 - 9)}{16}$

Step 3.5.1.2

Subtract $9$ from $3$ .

$f (3) = \frac{2 \cdot - 6}{16}$

Step 3.5.2

Reduce the expression by cancelling the common factors.

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Step 3.5.2.1

Multiply $2$ by $- 6$ .

$f (3) = \frac{- 12}{16}$

Step 3.5.2.2

Cancel the common factor of $- 12$ and $16$ .

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Step 3.5.2.2.1

Factor $4$ out of $- 12$ .

$f (3) = \frac{4 (- 3)}{16}$

Step 3.5.2.2.2

Cancel the common factors.

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Step 3.5.2.2.2.1

Factor $4$ out of $16$ .

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

Step 3.5.2.2.2.2

Cancel the common factor.

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

Step 3.5.2.2.2.3

Rewrite the expression.

$f (3) = \frac{- 3}{4}$

Step 3.5.2.3

Move the negative in front of the fraction.

$f (3) = - \frac{3}{4}$

Step 3.5.3

The final answer is $- \frac{3}{4}$ .

$- \frac{3}{4}$

Step 3.6

The $y$ value at $x = 3$ is $- \frac{3}{4}$ .

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

Step 3.7

Replace the variable $x$ with $6$ in the expression.

$f (6) = \frac{((6) - 1) ((6) - 9)}{16}$

Step 3.8

Simplify the result.

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Step 3.8.1

Simplify the numerator.

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Step 3.8.1.1

Subtract $1$ from $6$ .

$f (6) = \frac{5 (6 - 9)}{16}$

Step 3.8.1.2

Subtract $9$ from $6$ .

$f (6) = \frac{5 \cdot - 3}{16}$

Step 3.8.2

Simplify the expression.

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Step 3.8.2.1

Multiply $5$ by $- 3$ .

$f (6) = \frac{- 15}{16}$

Step 3.8.2.2

Move the negative in front of the fraction.

$f (6) = - \frac{15}{16}$

Step 3.8.3

The final answer is $- \frac{15}{16}$ .

$- \frac{15}{16}$

Step 3.9

The $y$ value at $x = 6$ is $- \frac{15}{16}$ .

$y = - \frac{15}{16}$

Step 3.10

Replace the variable $x$ with $7$ in the expression.

$f (7) = \frac{((7) - 1) ((7) - 9)}{16}$

Step 3.11

Simplify the result.

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Step 3.11.1

Simplify the numerator.

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Step 3.11.1.1

Subtract $1$ from $7$ .

$f (7) = \frac{6 (7 - 9)}{16}$

Step 3.11.1.2

Subtract $9$ from $7$ .

$f (7) = \frac{6 \cdot - 2}{16}$

Step 3.11.2

Reduce the expression by cancelling the common factors.

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Step 3.11.2.1

Multiply $6$ by $- 2$ .

$f (7) = \frac{- 12}{16}$

Step 3.11.2.2

Cancel the common factor of $- 12$ and $16$ .

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Step 3.11.2.2.1

Factor $4$ out of $- 12$ .

$f (7) = \frac{4 (- 3)}{16}$

Step 3.11.2.2.2

Cancel the common factors.

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Step 3.11.2.2.2.1

Factor $4$ out of $16$ .

$f (7) = \frac{4 \cdot - 3}{4 \cdot 4}$

Step 3.11.2.2.2.2

Cancel the common factor.

$f (7) = \frac{4 \cdot - 3}{4 \cdot 4}$

Step 3.11.2.2.2.3

Rewrite the expression.

$f (7) = \frac{- 3}{4}$

Step 3.11.2.3

Move the negative in front of the fraction.

$f (7) = - \frac{3}{4}$

Step 3.11.3

The final answer is $- \frac{3}{4}$ .

$- \frac{3}{4}$

Step 3.12

The $y$ value at $x = 7$ is $- \frac{3}{4}$ .

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

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 3 & - \frac{3}{4} \\ 4 & - \frac{15}{16} \\ 5 & - 1 \\ 6 & - \frac{15}{16} \\ 7 & - \frac{3}{4} \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(5, - 1)$

Focus: $(5, 3)$

Axis of Symmetry: $x = 5$

Directrix: $y = - 5$

$\begin{array}{c} x & y \\ 3 & - \frac{3}{4} \\ 4 & - \frac{15}{16} \\ 5 & - 1 \\ 6 & - \frac{15}{16} \\ 7 & - \frac{3}{4} \end{array}$

Step 5