Precalculus Examples

${(x - 1)}^{2} + 8 (y + 2) = 0$

Step 1

Solve for $y$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

${(x - 1)}^{2} + 8 y + 8 \cdot 2 = 0$

Step 1.1.2

Multiply $8$ by $2$ .

${(x - 1)}^{2} + 8 y + 16 = 0$

Step 1.2

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

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Step 1.2.1

Subtract ${(x - 1)}^{2}$ from both sides of the equation.

$8 y + 16 = - {(x - 1)}^{2}$

Step 1.2.2

Subtract $16$ from both sides of the equation.

$8 y = - {(x - 1)}^{2} - 16$

Step 1.3

Divide each term in $8 y = - {(x - 1)}^{2} - 16$ by $8$ and simplify.

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Step 1.3.1

Divide each term in $8 y = - {(x - 1)}^{2} - 16$ by $8$ .

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

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $8$ .

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Step 1.3.2.1.1

Cancel the common factor.

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

Step 1.3.2.1.2

Divide $y$ by $1$ .

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

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Move the negative in front of the fraction.

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

Step 1.3.3.1.2

Divide $- 16$ by $8$ .

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

Step 1.3.3.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

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

Step 1.3.3.3

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

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

Step 1.3.3.4

Combine the numerators over the common denominator.

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

Step 1.3.3.5

Multiply $- 2$ by $8$ .

$y = \frac{- {(x - 1)}^{2} - 16}{8}$

Step 1.3.3.6

Factor $- 1$ out of $- {(x - 1)}^{2}$ .

$y = \frac{- ({(x - 1)}^{2}) - 16}{8}$

Step 1.3.3.7

Rewrite $- 16$ as $- 1 (16)$ .

$y = \frac{- ({(x - 1)}^{2}) - 1 (16)}{8}$

Step 1.3.3.8

Factor $- 1$ out of $- ({(x - 1)}^{2}) - 1 (16)$ .

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

Step 1.3.3.9

Simplify the expression.

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Step 1.3.3.9.1

Rewrite $- ({(x - 1)}^{2} + 16)$ as $- 1 ({(x - 1)}^{2} + 16)$ .

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

Step 1.3.3.9.2

Move the negative in front of the fraction.

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

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

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

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Step 2.1.1.1

Negate $\frac{{(x - 1)}^{2} + 16}{8}$ .

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

Step 2.1.1.2

Reorder terms.

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

Step 2.1.2

Complete the square for $- \frac{1}{8} \cdot ({(x - 1)}^{2} + 16)$ .

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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 ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$- \frac{1}{8} \cdot ((x - 1) (x - 1) + 16)$

Step 2.1.2.1.1.2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 2.1.2.1.1.2.1

Apply the distributive property.

$- \frac{1}{8} \cdot (x (x - 1) - 1 (x - 1) + 16)$

Step 2.1.2.1.1.2.2

Apply the distributive property.

$- \frac{1}{8} \cdot (x \cdot x + x \cdot - 1 - 1 (x - 1) + 16)$

Step 2.1.2.1.1.2.3

Apply the distributive property.

$- \frac{1}{8} \cdot (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1 + 16)$

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 $x$ .

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

Step 2.1.2.1.1.3.1.2

Move $- 1$ to the left of $x$ .

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

Step 2.1.2.1.1.3.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 2.1.2.1.1.3.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 2.1.2.1.1.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 2.1.2.1.1.3.2

Subtract $x$ from $- x$ .

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

Step 2.1.2.1.2

Add $1$ and $16$ .

$- \frac{1}{8} \cdot (x^{2} - 2 x + 17)$

Step 2.1.2.1.3

Apply the distributive property.

$- \frac{1}{8} x^{2} - \frac{1}{8} (- 2 x) - \frac{1}{8} \cdot 17$

Step 2.1.2.1.4

Simplify.

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Step 2.1.2.1.4.1

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

$- \frac{x^{2}}{8} - \frac{1}{8} (- 2 x) - \frac{1}{8} \cdot 17$

Step 2.1.2.1.4.2

Cancel the common factor of $2$ .

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Step 2.1.2.1.4.2.1

Move the leading negative in $- \frac{1}{8}$ into the numerator.

$- \frac{x^{2}}{8} + \frac{- 1}{8} (- 2 x) - \frac{1}{8} \cdot 17$

Step 2.1.2.1.4.2.2

Factor $2$ out of $8$ .

$- \frac{x^{2}}{8} + \frac{- 1}{2 (4)} (- 2 x) - \frac{1}{8} \cdot 17$

Step 2.1.2.1.4.2.3

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

$- \frac{x^{2}}{8} + \frac{- 1}{2 (4)} (2 (- x)) - \frac{1}{8} \cdot 17$

Step 2.1.2.1.4.2.4

Cancel the common factor.

$- \frac{x^{2}}{8} + \frac{- 1}{2 \cdot 4} (2 (- x)) - \frac{1}{8} \cdot 17$

Step 2.1.2.1.4.2.5

Rewrite the expression.

$- \frac{x^{2}}{8} + \frac{- 1}{4} (- x) - \frac{1}{8} \cdot 17$

Step 2.1.2.1.4.3

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

$- \frac{x^{2}}{8} - \frac{- x}{4} - \frac{1}{8} \cdot 17$

Step 2.1.2.1.4.4

Multiply $- \frac{1}{8} \cdot 17$ .

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Step 2.1.2.1.4.4.1

Multiply $17$ by $- 1$ .

$- \frac{x^{2}}{8} - \frac{- x}{4} - 17 (\frac{1}{8})$

Step 2.1.2.1.4.4.2

Combine $- 17$ and $\frac{1}{8}$ .

$- \frac{x^{2}}{8} - \frac{- x}{4} + \frac{- 17}{8}$

Step 2.1.2.1.5

Simplify each term.

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Step 2.1.2.1.5.1

Move the negative in front of the fraction.

$- \frac{x^{2}}{8} - - \frac{x}{4} + \frac{- 17}{8}$

Step 2.1.2.1.5.2

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

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Step 2.1.2.1.5.2.1

Multiply $- 1$ by $- 1$ .

$- \frac{x^{2}}{8} + 1 \frac{x}{4} + \frac{- 17}{8}$

Step 2.1.2.1.5.2.2

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

$- \frac{x^{2}}{8} + \frac{x}{4} + \frac{- 17}{8}$

Step 2.1.2.1.5.3

Move the negative in front of the fraction.

$- \frac{x^{2}}{8} + \frac{x}{4} - \frac{17}{8}$

Step 2.1.2.2

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

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

$b = \frac{1}{4}$

$c = - \frac{17}{8}$

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

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

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.

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

Step 2.1.2.4.2.2

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

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Step 2.1.2.4.2.2.1

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

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

Step 2.1.2.4.2.2.2

Move the negative in front of the fraction.

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

Step 2.1.2.4.2.3

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

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

Step 2.1.2.4.2.4

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

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Step 2.1.2.4.2.4.1

Factor $2$ out of $2$ .

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

Step 2.1.2.4.2.4.2

Cancel the common factors.

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Step 2.1.2.4.2.4.2.1

Factor $2$ out of $8$ .

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

Step 2.1.2.4.2.4.2.2

Cancel the common factor.

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

Step 2.1.2.4.2.4.2.3

Rewrite the expression.

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

Step 2.1.2.4.2.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.1.2.4.2.6

Cancel the common factor of $4$ .

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Step 2.1.2.4.2.6.1

Factor $4$ out of $- (1 \cdot 4)$ .

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

Step 2.1.2.4.2.6.2

Cancel the common factor.

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

Step 2.1.2.4.2.6.3

Rewrite the expression.

$d = - 1 \cdot (1)$

Step 2.1.2.4.2.7

Multiply $- 1$ by $1$ .

$d = - 1$

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

$e = - \frac{17}{8} - \frac{{(\frac{1}{4})}^{2}}{4 (- \frac{1}{8})}$

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 $\frac{1}{4}$ .

$e = - \frac{17}{8} - \frac{\frac{1^{2}}{4^{2}}}{4 (- \frac{1}{8})}$

Step 2.1.2.5.2.1.1.2

One to any power is one.

$e = - \frac{17}{8} - \frac{\frac{1}{4^{2}}}{4 (- \frac{1}{8})}$

Step 2.1.2.5.2.1.1.3

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

$e = - \frac{17}{8} - \frac{\frac{1}{16}}{4 (- \frac{1}{8})}$

Step 2.1.2.5.2.1.2

Simplify the denominator.

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Step 2.1.2.5.2.1.2.1

Multiply $- 1$ by $4$ .

$e = - \frac{17}{8} - \frac{\frac{1}{16}}{- 4 (\frac{1}{8})}$

Step 2.1.2.5.2.1.2.2

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

$e = - \frac{17}{8} - \frac{\frac{1}{16}}{\frac{- 4}{8}}$

Step 2.1.2.5.2.1.3

Reduce the expression by cancelling the common factors.

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Step 2.1.2.5.2.1.3.1

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

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Step 2.1.2.5.2.1.3.1.1

Factor $4$ out of $- 4$ .

$e = - \frac{17}{8} - \frac{\frac{1}{16}}{\frac{4 (- 1)}{8}}$

Step 2.1.2.5.2.1.3.1.2

Cancel the common factors.

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Step 2.1.2.5.2.1.3.1.2.1

Factor $4$ out of $8$ .

$e = - \frac{17}{8} - \frac{\frac{1}{16}}{\frac{4 \cdot - 1}{4 \cdot 2}}$

Step 2.1.2.5.2.1.3.1.2.2

Cancel the common factor.

$e = - \frac{17}{8} - \frac{\frac{1}{16}}{\frac{4 \cdot - 1}{4 \cdot 2}}$

Step 2.1.2.5.2.1.3.1.2.3

Rewrite the expression.

$e = - \frac{17}{8} - \frac{\frac{1}{16}}{\frac{- 1}{2}}$

Step 2.1.2.5.2.1.3.2

Move the negative in front of the fraction.

$e = - \frac{17}{8} - \frac{\frac{1}{16}}{- \frac{1}{2}}$

Step 2.1.2.5.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{17}{8} - (\frac{1}{16} (- 1 \cdot 2))$

Step 2.1.2.5.2.1.5

Cancel the common factor of $2$ .

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Step 2.1.2.5.2.1.5.1

Factor $2$ out of $16$ .

$e = - \frac{17}{8} - (\frac{1}{2 (8)} (- 1 \cdot 2))$

Step 2.1.2.5.2.1.5.2

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

$e = - \frac{17}{8} - (\frac{1}{2 (8)} (2 \cdot - 1))$

Step 2.1.2.5.2.1.5.3

Cancel the common factor.

$e = - \frac{17}{8} - (\frac{1}{2 \cdot 8} (2 \cdot - 1))$

Step 2.1.2.5.2.1.5.4

Rewrite the expression.

$e = - \frac{17}{8} - (\frac{1}{8} \cdot - 1)$

Step 2.1.2.5.2.1.6

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

$e = - \frac{17}{8} - \frac{- 1}{8}$

Step 2.1.2.5.2.1.7

Move the negative in front of the fraction.

$e = - \frac{17}{8} - - \frac{1}{8}$

Step 2.1.2.5.2.1.8

Multiply $- - \frac{1}{8}$ .

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Step 2.1.2.5.2.1.8.1

Multiply $- 1$ by $- 1$ .

$e = - \frac{17}{8} + 1 (\frac{1}{8})$

Step 2.1.2.5.2.1.8.2

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

$e = - \frac{17}{8} + \frac{1}{8}$

Step 2.1.2.5.2.2

Combine the numerators over the common denominator.

$e = \frac{- 17 + 1}{8}$

Step 2.1.2.5.2.3

Add $- 17$ and $1$ .

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

Step 2.1.2.5.2.4

Divide $- 16$ by $8$ .

$e = - 2$

Step 2.1.2.6

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

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

Step 2.1.3

Set $y$ equal to the new right side.

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

Step 2.2

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

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

$h = 1$

$k = - 2$

Step 2.3

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

Opens Down

Step 2.4

Find the vertex $(h, k)$ .

$(1, - 2)$

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

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

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Step 2.5.3.1.1

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

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

Step 2.5.3.1.2

Move the negative in front of the fraction.

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

Step 2.5.3.2

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

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

Step 2.5.3.3

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

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Step 2.5.3.3.1

Factor $4$ out of $4$ .

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

Step 2.5.3.3.2

Cancel the common factors.

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Step 2.5.3.3.2.1

Factor $4$ out of $8$ .

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

Step 2.5.3.3.2.2

Cancel the common factor.

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

Step 2.5.3.3.2.3

Rewrite the expression.

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

Step 2.5.3.4

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 2)$

Step 2.5.3.5

Multiply $- (1 \cdot 2)$ .

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Step 2.5.3.5.1

Multiply $2$ by $1$ .

$- 1 \cdot 2$

Step 2.5.3.5.2

Multiply $- 1$ by $2$ .

$- 2$

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.

$(1, - 4)$

Step 2.7

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

$x = 1$

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.

$y = 0$

Step 2.9

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

Direction: Opens Down

Vertex: $(1, - 2)$

Focus: $(1, - 4)$

Axis of Symmetry: $x = 1$

Directrix: $y = 0$

Direction: Opens Down

Vertex: $(1, - 2)$

Focus: $(1, - 4)$

Axis of Symmetry: $x = 1$

Directrix: $y = 0$

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 $0$ in the expression.

$f (0) = - \frac{{(0)}^{2} - 2 \cdot 0 + 17}{8}$

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

Raising $0$ to any positive power yields $0$ .

$f (0) = - \frac{0 - 2 \cdot 0 + 17}{8}$

Step 3.2.1.2

Multiply $- 2$ by $0$ .

$f (0) = - \frac{0 + 0 + 17}{8}$

Step 3.2.1.3

Add $0$ and $0$ .

$f (0) = - \frac{0 + 17}{8}$

Step 3.2.1.4

Add $0$ and $17$ .

$f (0) = - \frac{17}{8}$

Step 3.2.2

The final answer is $- \frac{17}{8}$ .

$- \frac{17}{8}$

Step 3.3

The $y$ value at $x = 0$ is $- \frac{17}{8}$ .

$y = - \frac{17}{8}$

Step 3.4

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = - \frac{{(- 1)}^{2} - 2 \cdot - 1 + 17}{8}$

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

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

$f (- 1) = - \frac{1 - 2 \cdot - 1 + 17}{8}$

Step 3.5.1.2

Multiply $- 2$ by $- 1$ .

$f (- 1) = - \frac{1 + 2 + 17}{8}$

Step 3.5.1.3

Add $1$ and $2$ .

$f (- 1) = - \frac{3 + 17}{8}$

Step 3.5.1.4

Add $3$ and $17$ .

$f (- 1) = - \frac{20}{8}$

Step 3.5.2

Cancel the common factor of $20$ and $8$ .

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Step 3.5.2.1

Factor $4$ out of $20$ .

$f (- 1) = - \frac{4 (5)}{8}$

Step 3.5.2.2

Cancel the common factors.

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Step 3.5.2.2.1

Factor $4$ out of $8$ .

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

Step 3.5.2.2.2

Cancel the common factor.

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

Step 3.5.2.2.3

Rewrite the expression.

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

Step 3.5.3

The final answer is $- \frac{5}{2}$ .

$- \frac{5}{2}$

Step 3.6

The $y$ value at $x = - 1$ is $- \frac{5}{2}$ .

$y = - \frac{5}{2}$

Step 3.7

Replace the variable $x$ with $2$ in the expression.

$f (2) = - \frac{{(2)}^{2} - 2 \cdot 2 + 17}{8}$

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

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

$f (2) = - \frac{4 - 2 \cdot 2 + 17}{8}$

Step 3.8.1.2

Multiply $- 2$ by $2$ .

$f (2) = - \frac{4 - 4 + 17}{8}$

Step 3.8.1.3

Subtract $4$ from $4$ .

$f (2) = - \frac{0 + 17}{8}$

Step 3.8.1.4

Add $0$ and $17$ .

$f (2) = - \frac{17}{8}$

Step 3.8.2

The final answer is $- \frac{17}{8}$ .

$- \frac{17}{8}$

Step 3.9

The $y$ value at $x = 2$ is $- \frac{17}{8}$ .

$y = - \frac{17}{8}$

Step 3.10

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

$f (3) = - \frac{{(3)}^{2} - 2 \cdot 3 + 17}{8}$

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

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

$f (3) = - \frac{9 - 2 \cdot 3 + 17}{8}$

Step 3.11.1.2

Multiply $- 2$ by $3$ .

$f (3) = - \frac{9 - 6 + 17}{8}$

Step 3.11.1.3

Subtract $6$ from $9$ .

$f (3) = - \frac{3 + 17}{8}$

Step 3.11.1.4

Add $3$ and $17$ .

$f (3) = - \frac{20}{8}$

Step 3.11.2

Cancel the common factor of $20$ and $8$ .

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Step 3.11.2.1

Factor $4$ out of $20$ .

$f (3) = - \frac{4 (5)}{8}$

Step 3.11.2.2

Cancel the common factors.

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Step 3.11.2.2.1

Factor $4$ out of $8$ .

$f (3) = - \frac{4 \cdot 5}{4 \cdot 2}$

Step 3.11.2.2.2

Cancel the common factor.

$f (3) = - \frac{4 \cdot 5}{4 \cdot 2}$

Step 3.11.2.2.3

Rewrite the expression.

$f (3) = - \frac{5}{2}$

Step 3.11.3

The final answer is $- \frac{5}{2}$ .

$- \frac{5}{2}$

Step 3.12

The $y$ value at $x = 3$ is $- \frac{5}{2}$ .

$y = - \frac{5}{2}$

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & - \frac{5}{2} \\ 0 & - \frac{17}{8} \\ 1 & - 2 \\ 2 & - \frac{17}{8} \\ 3 & - \frac{5}{2} \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(1, - 2)$

Focus: $(1, - 4)$

Axis of Symmetry: $x = 1$

Directrix: $y = 0$

$\begin{array}{c} x & y \\ - 1 & - \frac{5}{2} \\ 0 & - \frac{17}{8} \\ 1 & - 2 \\ 2 & - \frac{17}{8} \\ 3 & - \frac{5}{2} \end{array}$

Step 5