Precalculus Examples

x^{2} - 8 x - 8 y - 16 = 0

$x^{2} - 8 x - 8 y - 16 = 0$

Step 1

Solve for

y

$y$ .

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Step 1.1

Move all terms not containing

y

$y$ to the right side of the equation.

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Step 1.1.1

Subtract

x^{2}

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

- 8 x - 8 y - 16 = - x^{2}

$- 8 x - 8 y - 16 = - x^{2}$

Step 1.1.2

Add

8 x

$8 x$ to both sides of the equation.

- 8 y - 16 = - x^{2} + 8 x

$- 8 y - 16 = - x^{2} + 8 x$

Step 1.1.3

Add

16

$16$ to both sides of the equation.

- 8 y = - x^{2} + 8 x + 16

$- 8 y = - x^{2} + 8 x + 16$

- 8 y = - x^{2} + 8 x + 16

$- 8 y = - x^{2} + 8 x + 16$

Step 1.2

Divide each term in

- 8 y = - x^{2} + 8 x + 16

$- 8 y = - x^{2} + 8 x + 16$ by

- 8

$- 8$ and simplify.

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Step 1.2.1

Divide each term in

- 8 y = - x^{2} + 8 x + 16

$- 8 y = - x^{2} + 8 x + 16$ by

- 8

$- 8$ .

\frac{- 8 y}{- 8} = \frac{- x^{2}}{- 8} + \frac{8 x}{- 8} + \frac{16}{- 8}

$\frac{- 8 y}{- 8} = \frac{- x^{2}}{- 8} + \frac{8 x}{- 8} + \frac{16}{- 8}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of

- 8

$- 8$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 8 y}{- 8} = \frac{- x^{2}}{- 8} + \frac{8 x}{- 8} + \frac{16}{- 8}$

Step 1.2.2.1.2

Divide

$y$ by

$1$ .

$y = \frac{- x^{2}}{- 8} + \frac{8 x}{- 8} + \frac{16}{- 8}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Dividing two negative values results in a positive value.

$y = \frac{x^{2}}{8} + \frac{8 x}{- 8} + \frac{16}{- 8}$

Step 1.2.3.1.2

Cancel the common factor of

$8$ and

$- 8$ .

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Step 1.2.3.1.2.1

Factor

$8$ out of

$8 x$ .

$y = \frac{x^{2}}{8} + \frac{8 (x)}{- 8} + \frac{16}{- 8}$

Step 1.2.3.1.2.2

Move the negative one from the denominator of

$\frac{x}{- 1}$ .

$y = \frac{x^{2}}{8} - 1 \cdot x + \frac{16}{- 8}$

Step 1.2.3.1.3

Rewrite

$- 1 \cdot x$ as

$- x$ .

$y = \frac{x^{2}}{8} - x + \frac{16}{- 8}$

Step 1.2.3.1.4

Divide

$16$ by

$- 8$ .

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

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

Complete the square for

$\frac{x^{2}}{8} - x - 2$ .

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Step 2.1.1.1

Use the form

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

$a$ ,

$b$ , and

$c$ .

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

$b = - 1$

$c = - 2$

Step 2.1.1.2

Consider the vertex form of a parabola.

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

Step 2.1.1.3

Find the value of

$d$ using the formula

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

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Step 2.1.1.3.1

Substitute the values of

$a$ and

$b$ into the formula

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

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

Step 2.1.1.3.2

Simplify the right side.

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Step 2.1.1.3.2.1

Combine

$2$ and

$\frac{1}{8}$ .

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

Step 2.1.1.3.2.2

Cancel the common factor of

$2$ and

$8$ .

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Step 2.1.1.3.2.2.1

Factor

$2$ out of

$2$ .

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

Step 2.1.1.3.2.2.2

Cancel the common factors.

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Step 2.1.1.3.2.2.2.1

Factor

$2$ out of

$8$ .

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

Step 2.1.1.3.2.2.2.2

Cancel the common factor.

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

Step 2.1.1.3.2.2.2.3

Rewrite the expression.

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

Step 2.1.1.3.2.3

Multiply the numerator by the reciprocal of the denominator.

$d = - 1 \cdot 4$

Step 2.1.1.3.2.4

Multiply

$- 1$ by

$4$ .

$d = - 4$

Step 2.1.1.4

Find the value of

$e$ using the formula

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

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Step 2.1.1.4.1

Substitute the values of

$c$ ,

$b$ and

$a$ into the formula

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

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

Step 2.1.1.4.2

Simplify the right side.

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Step 2.1.1.4.2.1

Simplify each term.

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Step 2.1.1.4.2.1.1

Raise

$- 1$ to the power of

$2$ .

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

Step 2.1.1.4.2.1.2

Combine

$4$ and

$\frac{1}{8}$ .

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

Step 2.1.1.4.2.1.3

Cancel the common factor of

$4$ and

$8$ .

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Step 2.1.1.4.2.1.3.1

Factor

$4$ out of

$4$ .

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

Step 2.1.1.4.2.1.3.2

Cancel the common factors.

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Step 2.1.1.4.2.1.3.2.1

Factor

$4$ out of

$8$ .

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

Step 2.1.1.4.2.1.3.2.2

Cancel the common factor.

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

Step 2.1.1.4.2.1.3.2.3

Rewrite the expression.

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

Step 2.1.1.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = - 2 - (1 \cdot 2)$

Step 2.1.1.4.2.1.5

Multiply

$- (1 \cdot 2)$ .

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Step 2.1.1.4.2.1.5.1

Multiply

$2$ by

$1$ .

$e = - 2 - 1 \cdot 2$

Step 2.1.1.4.2.1.5.2

Multiply

$- 1$ by

$2$ .

$e = - 2 - 2$

Step 2.1.1.4.2.2

Subtract

$2$ from

$- 2$ .

$e = - 4$

Step 2.1.1.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

$\frac{1}{8} {(x - 4)}^{2} - 4$ .

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

Step 2.1.2

Set

$y$ equal to the new right side.

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

Step 2.2

Use the vertex form,

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

$a$ ,

$h$ , and

$k$ .

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

$h = 4$

$k = - 4$

Step 2.3

Since the value of

$a$ is positive, the parabola opens up.

Opens Up

Step 2.4

Find the vertex

$(h, k)$ .

$(4, - 4)$

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.

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

Step 2.5.3

Simplify.

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Step 2.5.3.1

Combine

$4$ and

$\frac{1}{8}$ .

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

Step 2.5.3.2

Cancel the common factor of

$4$ and

$8$ .

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Step 2.5.3.2.1

Factor

$4$ out of

$4$ .

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

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

$8$ .

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

Step 2.5.3.2.2.2

Cancel the common factor.

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

Step 2.5.3.2.2.3

Rewrite the expression.

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

Step 2.5.3.3

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 2$

Step 2.5.3.4

Multiply

$2$ by

$1$ .

$2$

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.

$(4, - 2)$

Step 2.7

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

$x = 4$

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.

$y = - 6$

Step 2.9

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

Direction: Opens Up

Vertex:

$(4, - 4)$

Focus:

$(4, - 2)$

Axis of Symmetry:

$x = 4$

Directrix:

$y = - 6$

Direction: Opens Up

Vertex:

$(4, - 4)$

Focus:

$(4, - 2)$

Axis of Symmetry:

$x = 4$

Directrix:

$y = - 6$

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

$3$ in the expression.

$f (3) = \frac{{(3)}^{2}}{8} - (3) - 2$

Step 3.2

Simplify the result.

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Step 3.2.1

Find the common denominator.

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Step 3.2.1.1

Write

$- (3)$ as a fraction with denominator

$1$ .

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

Step 3.2.1.2

Multiply

$\frac{- (3)}{1}$ by

$\frac{8}{8}$ .

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

Step 3.2.1.3

Multiply

$\frac{- (3)}{1}$ by

$\frac{8}{8}$ .

$f (3) = \frac{{(3)}^{2}}{8} + \frac{- 3 \cdot 8}{8} - 2$

Step 3.2.1.4

Write

$- 2$ as a fraction with denominator

$1$ .

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

Step 3.2.1.5

Multiply

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

$\frac{8}{8}$ .

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

Step 3.2.1.6

Multiply

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

$\frac{8}{8}$ .

$f (3) = \frac{{(3)}^{2}}{8} + \frac{- 3 \cdot 8}{8} + \frac{- 2 \cdot 8}{8}$

Step 3.2.2

Combine the numerators over the common denominator.

$f (3) = \frac{{(3)}^{2} - 3 \cdot 8 - 2 \cdot 8}{8}$

Step 3.2.3

Simplify each term.

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Step 3.2.3.1

Raise

$3$ to the power of

$2$ .

$f (3) = \frac{9 - 3 \cdot 8 - 2 \cdot 8}{8}$

Step 3.2.3.2

Multiply

$- (3) \cdot 8$ .

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Step 3.2.3.2.1

Multiply

$- 1$ by

$3$ .

$f (3) = \frac{9 - 3 \cdot 8 - 2 \cdot 8}{8}$

Step 3.2.3.2.2

Multiply

$- 3$ by

$8$ .

$f (3) = \frac{9 - 24 - 2 \cdot 8}{8}$

Step 3.2.3.3

Multiply

$- 2$ by

$8$ .

$f (3) = \frac{9 - 24 - 16}{8}$

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Subtract

$24$ from

$9$ .

$f (3) = \frac{- 15 - 16}{8}$

Step 3.2.4.2

Subtract

$16$ from

$- 15$ .

$f (3) = \frac{- 31}{8}$

Step 3.2.4.3

Move the negative in front of the fraction.

$f (3) = - \frac{31}{8}$

Step 3.2.5

The final answer is

$- \frac{31}{8}$ .

$- \frac{31}{8}$

Step 3.3

The

$y$ value at

$x = 3$ is

$- \frac{31}{8}$ .

$y = - \frac{31}{8}$

Step 3.4

Replace the variable

$x$ with

$2$ in the expression.

$f (2) = \frac{{(2)}^{2}}{8} - (2) - 2$

Step 3.5

Simplify the result.

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Step 3.5.1

Find the common denominator.

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Step 3.5.1.1

Write

$- (2)$ as a fraction with denominator

$1$ .

$f (2) = \frac{{(2)}^{2}}{8} + \frac{- (2)}{1} - 2$

Step 3.5.1.2

Multiply

$\frac{- (2)}{1}$ by

$\frac{8}{8}$ .

$f (2) = \frac{{(2)}^{2}}{8} + \frac{- (2)}{1} \cdot \frac{8}{8} - 2$

Step 3.5.1.3

Multiply

$\frac{- (2)}{1}$ by

$\frac{8}{8}$ .

$f (2) = \frac{{(2)}^{2}}{8} + \frac{- 2 \cdot 8}{8} - 2$

Step 3.5.1.4

Write

$- 2$ as a fraction with denominator

$1$ .

$f (2) = \frac{{(2)}^{2}}{8} + \frac{- 2 \cdot 8}{8} + \frac{- 2}{1}$

Step 3.5.1.5

Multiply

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

$\frac{8}{8}$ .

$f (2) = \frac{{(2)}^{2}}{8} + \frac{- 2 \cdot 8}{8} + \frac{- 2}{1} \cdot \frac{8}{8}$

Step 3.5.1.6

Multiply

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

$\frac{8}{8}$ .

$f (2) = \frac{{(2)}^{2}}{8} + \frac{- 2 \cdot 8}{8} + \frac{- 2 \cdot 8}{8}$

Step 3.5.2

Combine the numerators over the common denominator.

$f (2) = \frac{{(2)}^{2} - 2 \cdot 8 - 2 \cdot 8}{8}$

Step 3.5.3

Simplify each term.

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Step 3.5.3.1

Raise

$2$ to the power of

$2$ .

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

Step 3.5.3.2

Multiply

$- (2) \cdot 8$ .

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Step 3.5.3.2.1

Multiply

$- 1$ by

$2$ .

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

Step 3.5.3.2.2

Multiply

$- 2$ by

$8$ .

$f (2) = \frac{4 - 16 - 2 \cdot 8}{8}$

Step 3.5.3.3

Multiply

$- 2$ by

$8$ .

$f (2) = \frac{4 - 16 - 16}{8}$

Step 3.5.4

Reduce the expression by cancelling the common factors.

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Step 3.5.4.1

Subtract

$16$ from

$4$ .

$f (2) = \frac{- 12 - 16}{8}$

Step 3.5.4.2

Subtract

$16$ from

$- 12$ .

$f (2) = \frac{- 28}{8}$

Step 3.5.4.3

Cancel the common factor of

$- 28$ and

$8$ .

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Step 3.5.4.3.1

Factor

$4$ out of

$- 28$ .

$f (2) = \frac{4 (- 7)}{8}$

Step 3.5.4.3.2

Cancel the common factors.

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Step 3.5.4.3.2.1

Factor

$4$ out of

$8$ .

$f (2) = \frac{4 \cdot - 7}{4 \cdot 2}$

Step 3.5.4.3.2.2

Cancel the common factor.

$f (2) = \frac{4 \cdot - 7}{4 \cdot 2}$

Step 3.5.4.3.2.3

Rewrite the expression.

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

Step 3.5.4.4

Move the negative in front of the fraction.

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

Step 3.5.5

The final answer is

$- \frac{7}{2}$ .

$- \frac{7}{2}$

Step 3.6

The

$y$ value at

$x = 2$ is

$- \frac{7}{2}$ .

$y = - \frac{7}{2}$

Step 3.7

Replace the variable

$x$ with

$5$ in the expression.

$f (5) = \frac{{(5)}^{2}}{8} - (5) - 2$

Step 3.8

Simplify the result.

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Step 3.8.1

Find the common denominator.

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Step 3.8.1.1

Write

$- (5)$ as a fraction with denominator

$1$ .

$f (5) = \frac{{(5)}^{2}}{8} + \frac{- (5)}{1} - 2$

Step 3.8.1.2

Multiply

$\frac{- (5)}{1}$ by

$\frac{8}{8}$ .

$f (5) = \frac{{(5)}^{2}}{8} + \frac{- (5)}{1} \cdot \frac{8}{8} - 2$

Step 3.8.1.3

Multiply

$\frac{- (5)}{1}$ by

$\frac{8}{8}$ .

$f (5) = \frac{{(5)}^{2}}{8} + \frac{- 5 \cdot 8}{8} - 2$

Step 3.8.1.4

Write

$- 2$ as a fraction with denominator

$1$ .

$f (5) = \frac{{(5)}^{2}}{8} + \frac{- 5 \cdot 8}{8} + \frac{- 2}{1}$

Step 3.8.1.5

Multiply

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

$\frac{8}{8}$ .

$f (5) = \frac{{(5)}^{2}}{8} + \frac{- 5 \cdot 8}{8} + \frac{- 2}{1} \cdot \frac{8}{8}$

Step 3.8.1.6

Multiply

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

$\frac{8}{8}$ .

$f (5) = \frac{{(5)}^{2}}{8} + \frac{- 5 \cdot 8}{8} + \frac{- 2 \cdot 8}{8}$

Step 3.8.2

Combine the numerators over the common denominator.

$f (5) = \frac{{(5)}^{2} - 5 \cdot 8 - 2 \cdot 8}{8}$

Step 3.8.3

Simplify each term.

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Step 3.8.3.1

Raise

$5$ to the power of

$2$ .

$f (5) = \frac{25 - 5 \cdot 8 - 2 \cdot 8}{8}$

Step 3.8.3.2

Multiply

$- (5) \cdot 8$ .

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Step 3.8.3.2.1

Multiply

$- 1$ by

$5$ .

$f (5) = \frac{25 - 5 \cdot 8 - 2 \cdot 8}{8}$

Step 3.8.3.2.2

Multiply

$- 5$ by

$8$ .

$f (5) = \frac{25 - 40 - 2 \cdot 8}{8}$

Step 3.8.3.3

Multiply

$- 2$ by

$8$ .

$f (5) = \frac{25 - 40 - 16}{8}$

Step 3.8.4

Simplify the expression.

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Step 3.8.4.1

Subtract

$40$ from

$25$ .

$f (5) = \frac{- 15 - 16}{8}$

Step 3.8.4.2

Subtract

$16$ from

$- 15$ .

$f (5) = \frac{- 31}{8}$

Step 3.8.4.3

Move the negative in front of the fraction.

$f (5) = - \frac{31}{8}$

Step 3.8.5

The final answer is

$- \frac{31}{8}$ .

$- \frac{31}{8}$

Step 3.9

The

$y$ value at

$x = 5$ is

$- \frac{31}{8}$ .

$y = - \frac{31}{8}$

Step 3.10

Replace the variable

$x$ with

$6$ in the expression.

$f (6) = \frac{{(6)}^{2}}{8} - (6) - 2$

Step 3.11

Simplify the result.

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Step 3.11.1

Find the common denominator.

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Step 3.11.1.1

Write

$- (6)$ as a fraction with denominator

$1$ .

$f (6) = \frac{{(6)}^{2}}{8} + \frac{- (6)}{1} - 2$

Step 3.11.1.2

Multiply

$\frac{- (6)}{1}$ by

$\frac{8}{8}$ .

$f (6) = \frac{{(6)}^{2}}{8} + \frac{- (6)}{1} \cdot \frac{8}{8} - 2$

Step 3.11.1.3

Multiply

$\frac{- (6)}{1}$ by

$\frac{8}{8}$ .

$f (6) = \frac{{(6)}^{2}}{8} + \frac{- 6 \cdot 8}{8} - 2$

Step 3.11.1.4

Write

$- 2$ as a fraction with denominator

$1$ .

$f (6) = \frac{{(6)}^{2}}{8} + \frac{- 6 \cdot 8}{8} + \frac{- 2}{1}$

Step 3.11.1.5

Multiply

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

$\frac{8}{8}$ .

$f (6) = \frac{{(6)}^{2}}{8} + \frac{- 6 \cdot 8}{8} + \frac{- 2}{1} \cdot \frac{8}{8}$

Step 3.11.1.6

Multiply

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

$\frac{8}{8}$ .

$f (6) = \frac{{(6)}^{2}}{8} + \frac{- 6 \cdot 8}{8} + \frac{- 2 \cdot 8}{8}$

Step 3.11.2

Combine the numerators over the common denominator.

$f (6) = \frac{{(6)}^{2} - 6 \cdot 8 - 2 \cdot 8}{8}$

Step 3.11.3

Simplify each term.

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Step 3.11.3.1

Raise

$6$ to the power of

$2$ .

$f (6) = \frac{36 - 6 \cdot 8 - 2 \cdot 8}{8}$

Step 3.11.3.2

Multiply

$- (6) \cdot 8$ .

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Step 3.11.3.2.1

Multiply

$- 1$ by

$6$ .

$f (6) = \frac{36 - 6 \cdot 8 - 2 \cdot 8}{8}$

Step 3.11.3.2.2

Multiply

$- 6$ by

$8$ .

$f (6) = \frac{36 - 48 - 2 \cdot 8}{8}$

Step 3.11.3.3

Multiply

$- 2$ by

$8$ .

$f (6) = \frac{36 - 48 - 16}{8}$

Step 3.11.4

Reduce the expression by cancelling the common factors.

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Step 3.11.4.1

Subtract

$48$ from

$36$ .

$f (6) = \frac{- 12 - 16}{8}$

Step 3.11.4.2

Subtract

$16$ from

$- 12$ .

$f (6) = \frac{- 28}{8}$

Step 3.11.4.3

Cancel the common factor of

$- 28$ and

$8$ .

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Step 3.11.4.3.1

Factor

$4$ out of

$- 28$ .

$f (6) = \frac{4 (- 7)}{8}$

Step 3.11.4.3.2

Cancel the common factors.

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Step 3.11.4.3.2.1

Factor

$4$ out of

$8$ .

$f (6) = \frac{4 \cdot - 7}{4 \cdot 2}$

Step 3.11.4.3.2.2

Cancel the common factor.

$f (6) = \frac{4 \cdot - 7}{4 \cdot 2}$

Step 3.11.4.3.2.3

Rewrite the expression.

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

Step 3.11.4.4

Move the negative in front of the fraction.

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

Step 3.11.5

The final answer is

$- \frac{7}{2}$ .

$- \frac{7}{2}$

Step 3.12

The

$y$ value at

$x = 6$ is

$- \frac{7}{2}$ .

$y = - \frac{7}{2}$

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 2 & - \frac{7}{2} \\ 3 & - \frac{31}{8} \\ 4 & - 4 \\ 5 & - \frac{31}{8} \\ 6 & - \frac{7}{2} \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex:

$(4, - 4)$

Focus:

$(4, - 2)$

Axis of Symmetry:

$x = 4$

Directrix:

$y = - 6$

$\begin{array}{c} x & y \\ 2 & - \frac{7}{2} \\ 3 & - \frac{31}{8} \\ 4 & - 4 \\ 5 & - \frac{31}{8} \\ 6 & - \frac{7}{2} \end{array}$

Step 5