Pre-Algebra Examples

$y - 5 = \frac{1}{8} \cdot {(x + 3)}^{2}$

Step 1

Solve for $y$ .

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Step 1.1

Simplify $\frac{1}{8} \cdot {(x + 3)}^{2}$ .

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Step 1.1.1

Rewrite.

$y - 5 = 0 + 0 + \frac{1}{8} \cdot {(x + 3)}^{2}$

Step 1.1.2

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$y - 5 = \frac{1}{8} \cdot ((x + 3) (x + 3))$

Step 1.1.3

Expand $(x + 3) (x + 3)$ using the FOIL Method.

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Step 1.1.3.1

Apply the distributive property.

$y - 5 = \frac{1}{8} \cdot (x (x + 3) + 3 (x + 3))$

Step 1.1.3.2

Apply the distributive property.

$y - 5 = \frac{1}{8} \cdot (x \cdot x + x \cdot 3 + 3 (x + 3))$

Step 1.1.3.3

Apply the distributive property.

$y - 5 = \frac{1}{8} \cdot (x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3)$

Step 1.1.4

Simplify and combine like terms.

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Step 1.1.4.1

Simplify each term.

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Step 1.1.4.1.1

Multiply $x$ by $x$ .

$y - 5 = \frac{1}{8} \cdot (x^{2} + x \cdot 3 + 3 x + 3 \cdot 3)$

Step 1.1.4.1.2

Move $3$ to the left of $x$ .

$y - 5 = \frac{1}{8} \cdot (x^{2} + 3 \cdot x + 3 x + 3 \cdot 3)$

Step 1.1.4.1.3

Multiply $3$ by $3$ .

$y - 5 = \frac{1}{8} \cdot (x^{2} + 3 x + 3 x + 9)$

Step 1.1.4.2

Add $3 x$ and $3 x$ .

$y - 5 = \frac{1}{8} \cdot (x^{2} + 6 x + 9)$

Step 1.1.5

Apply the distributive property.

$y - 5 = \frac{1}{8} x^{2} + \frac{1}{8} (6 x) + \frac{1}{8} \cdot 9$

Step 1.1.6

Simplify.

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Step 1.1.6.1

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

$y - 5 = \frac{x^{2}}{8} + \frac{1}{8} (6 x) + \frac{1}{8} \cdot 9$

Step 1.1.6.2

Cancel the common factor of $2$ .

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Step 1.1.6.2.1

Factor $2$ out of $8$ .

$y - 5 = \frac{x^{2}}{8} + \frac{1}{2 (4)} (6 x) + \frac{1}{8} \cdot 9$

Step 1.1.6.2.2

Factor $2$ out of $6 x$ .

$y - 5 = \frac{x^{2}}{8} + \frac{1}{2 (4)} (2 (3 x)) + \frac{1}{8} \cdot 9$

Step 1.1.6.2.3

Cancel the common factor.

$y - 5 = \frac{x^{2}}{8} + \frac{1}{2 \cdot 4} (2 (3 x)) + \frac{1}{8} \cdot 9$

Step 1.1.6.2.4

Rewrite the expression.

$y - 5 = \frac{x^{2}}{8} + \frac{1}{4} (3 x) + \frac{1}{8} \cdot 9$

Step 1.1.6.3

Combine $3$ and $\frac{1}{4}$ .

$y - 5 = \frac{x^{2}}{8} + \frac{3}{4} x + \frac{1}{8} \cdot 9$

Step 1.1.6.4

Combine $\frac{3}{4}$ and $x$ .

$y - 5 = \frac{x^{2}}{8} + \frac{3 x}{4} + \frac{1}{8} \cdot 9$

Step 1.1.6.5

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

$y - 5 = \frac{x^{2}}{8} + \frac{3 x}{4} + \frac{9}{8}$

Step 1.2

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

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Step 1.2.1

Add $5$ to both sides of the equation.

$y = \frac{x^{2}}{8} + \frac{3 x}{4} + \frac{9}{8} + 5$

Step 1.2.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y = \frac{x^{2}}{8} + \frac{3 x}{4} + \frac{9}{8} + 5 \cdot \frac{8}{8}$

Step 1.2.3

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

$y = \frac{x^{2}}{8} + \frac{3 x}{4} + \frac{9}{8} + \frac{5 \cdot 8}{8}$

Step 1.2.4

Combine the numerators over the common denominator.

$y = \frac{x^{2}}{8} + \frac{3 x}{4} + \frac{9 + 5 \cdot 8}{8}$

Step 1.2.5

Simplify the numerator.

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Step 1.2.5.1

Multiply $5$ by $8$ .

$y = \frac{x^{2}}{8} + \frac{3 x}{4} + \frac{9 + 40}{8}$

Step 1.2.5.2

Add $9$ and $40$ .

$y = \frac{x^{2}}{8} + \frac{3 x}{4} + \frac{49}{8}$

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

Complete the square for $\frac{x^{2}}{8} + \frac{3 x}{4} + \frac{49}{8}$ .

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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 $c$ .

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

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

$c = \frac{49}{8}$

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

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

Step 2.1.1.3.2

Simplify the right side.

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Step 2.1.1.3.2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.1.1.3.2.2

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

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

Step 2.1.1.3.2.3

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

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Step 2.1.1.3.2.3.1

Factor $2$ out of $2$ .

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

Step 2.1.1.3.2.3.2

Cancel the common factors.

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Step 2.1.1.3.2.3.2.1

Factor $2$ out of $8$ .

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

Step 2.1.1.3.2.3.2.2

Cancel the common factor.

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

Step 2.1.1.3.2.3.2.3

Rewrite the expression.

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

Step 2.1.1.3.2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.1.1.3.2.5

Cancel the common factor of $4$ .

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Step 2.1.1.3.2.5.1

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

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

Step 2.1.1.3.2.5.2

Cancel the common factor.

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

Step 2.1.1.3.2.5.3

Rewrite the expression.

$d = 3$

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

$e = \frac{49}{8} - \frac{{(\frac{3}{4})}^{2}}{4 (\frac{1}{8})}$

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

Simplify the numerator.

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Step 2.1.1.4.2.1.1.1

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

$e = \frac{49}{8} - \frac{\frac{3^{2}}{4^{2}}}{4 (\frac{1}{8})}$

Step 2.1.1.4.2.1.1.2

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

$e = \frac{49}{8} - \frac{\frac{9}{4^{2}}}{4 (\frac{1}{8})}$

Step 2.1.1.4.2.1.1.3

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

$e = \frac{49}{8} - \frac{\frac{9}{16}}{4 (\frac{1}{8})}$

Step 2.1.1.4.2.1.2

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

$e = \frac{49}{8} - \frac{\frac{9}{16}}{\frac{4}{8}}$

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 $4$ .

$e = \frac{49}{8} - \frac{\frac{9}{16}}{\frac{4 (1)}{8}}$

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 $8$ .

$e = \frac{49}{8} - \frac{\frac{9}{16}}{\frac{4 \cdot 1}{4 \cdot 2}}$

Step 2.1.1.4.2.1.3.2.2

Cancel the common factor.

$e = \frac{49}{8} - \frac{\frac{9}{16}}{\frac{4 \cdot 1}{4 \cdot 2}}$

Step 2.1.1.4.2.1.3.2.3

Rewrite the expression.

$e = \frac{49}{8} - \frac{\frac{9}{16}}{\frac{1}{2}}$

Step 2.1.1.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{49}{8} - (\frac{9}{16} \cdot 2)$

Step 2.1.1.4.2.1.5

Cancel the common factor of $2$ .

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Step 2.1.1.4.2.1.5.1

Factor $2$ out of $16$ .

$e = \frac{49}{8} - (\frac{9}{2 (8)} \cdot 2)$

Step 2.1.1.4.2.1.5.2

Cancel the common factor.

$e = \frac{49}{8} - (\frac{9}{2 \cdot 8} \cdot 2)$

Step 2.1.1.4.2.1.5.3

Rewrite the expression.

$e = \frac{49}{8} - \frac{9}{8}$

Step 2.1.1.4.2.2

Combine the numerators over the common denominator.

$e = \frac{49 - 9}{8}$

Step 2.1.1.4.2.3

Subtract $9$ from $49$ .

$e = \frac{40}{8}$

Step 2.1.1.4.2.4

Divide $40$ by $8$ .

$e = 5$

Step 2.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{8} {(x + 3)}^{2} + 5$ .

$\frac{1}{8} {(x + 3)}^{2} + 5$

Step 2.1.2

Set $y$ equal to the new right side.

$y = \frac{1}{8} \cdot {(x + 3)}^{2} + 5$

Step 2.2

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

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

$h = - 3$

$k = 5$

Step 2.3

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

Opens Up

Step 2.4

Find the vertex $(h, k)$ .

$(- 3, 5)$

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.

$(- 3, 7)$

Step 2.7

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

$x = - 3$

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.

$y = 3$

Step 2.9

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

Direction: Opens Up

Vertex: $(- 3, 5)$

Focus: $(- 3, 7)$

Axis of Symmetry: $x = - 3$

Directrix: $y = 3$

Direction: Opens Up

Vertex: $(- 3, 5)$

Focus: $(- 3, 7)$

Axis of Symmetry: $x = - 3$

Directrix: $y = 3$

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.

$f (- 4) = \frac{{(- 4)}^{2}}{8} + \frac{3 (- 4)}{4} + \frac{49}{8}$

Step 3.2

Simplify the result.

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Step 3.2.1

Combine the numerators over the common denominator.

$f (- 4) = \frac{{(- 4)}^{2} + 49}{8} + \frac{3 (- 4)}{4}$

Step 3.2.2

Simplify the expression.

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Step 3.2.2.1

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

$f (- 4) = \frac{16 + 49}{8} + \frac{3 (- 4)}{4}$

Step 3.2.2.2

Add $16$ and $49$ .

$f (- 4) = \frac{65}{8} + \frac{3 (- 4)}{4}$

Step 3.2.2.3

Multiply $3$ by $- 4$ .

$f (- 4) = \frac{65}{8} + \frac{- 12}{4}$

Step 3.2.2.4

Divide $- 12$ by $4$ .

$f (- 4) = \frac{65}{8} - 3$

Step 3.2.3

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$f (- 4) = \frac{65}{8} - 3 \cdot \frac{8}{8}$

Step 3.2.4

Combine $- 3$ and $\frac{8}{8}$ .

$f (- 4) = \frac{65}{8} + \frac{- 3 \cdot 8}{8}$

Step 3.2.5

Combine the numerators over the common denominator.

$f (- 4) = \frac{65 - 3 \cdot 8}{8}$

Step 3.2.6

Simplify the numerator.

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Step 3.2.6.1

Multiply $- 3$ by $8$ .

$f (- 4) = \frac{65 - 24}{8}$

Step 3.2.6.2

Subtract $24$ from $65$ .

$f (- 4) = \frac{41}{8}$

Step 3.2.7

The final answer is $\frac{41}{8}$ .

$\frac{41}{8}$

Step 3.3

The $y$ value at $x = - 4$ is $\frac{41}{8}$ .

$y = \frac{41}{8}$

Step 3.4

Replace the variable $x$ with $- 5$ in the expression.

$f (- 5) = \frac{{(- 5)}^{2}}{8} + \frac{3 (- 5)}{4} + \frac{49}{8}$

Step 3.5

Simplify the result.

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Step 3.5.1

Combine fractions.

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Step 3.5.1.1

Combine the numerators over the common denominator.

$f (- 5) = \frac{{(- 5)}^{2} + 49}{8} + \frac{3 (- 5)}{4}$

Step 3.5.1.2

Simplify the expression.

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Step 3.5.1.2.1

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

$f (- 5) = \frac{25 + 49}{8} + \frac{3 (- 5)}{4}$

Step 3.5.1.2.2

Add $25$ and $49$ .

$f (- 5) = \frac{74}{8} + \frac{3 (- 5)}{4}$

Step 3.5.1.2.3

Multiply $3$ by $- 5$ .

$f (- 5) = \frac{74}{8} + \frac{- 15}{4}$

Step 3.5.2

Simplify each term.

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Step 3.5.2.1

Cancel the common factor of $74$ and $8$ .

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Step 3.5.2.1.1

Factor $2$ out of $74$ .

$f (- 5) = \frac{2 (37)}{8} + \frac{- 15}{4}$

Step 3.5.2.1.2

Cancel the common factors.

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Step 3.5.2.1.2.1

Factor $2$ out of $8$ .

$f (- 5) = \frac{2 \cdot 37}{2 \cdot 4} + \frac{- 15}{4}$

Step 3.5.2.1.2.2

Cancel the common factor.

$f (- 5) = \frac{2 \cdot 37}{2 \cdot 4} + \frac{- 15}{4}$

Step 3.5.2.1.2.3

Rewrite the expression.

$f (- 5) = \frac{37}{4} + \frac{- 15}{4}$

Step 3.5.2.2

Move the negative in front of the fraction.

$f (- 5) = \frac{37}{4} - \frac{15}{4}$

Step 3.5.3

Simplify terms.

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Step 3.5.3.1

Combine the numerators over the common denominator.

$f (- 5) = \frac{37 - 15}{4}$

Step 3.5.3.2

Subtract $15$ from $37$ .

$f (- 5) = \frac{22}{4}$

Step 3.5.3.3

Cancel the common factor of $22$ and $4$ .

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Step 3.5.3.3.1

Factor $2$ out of $22$ .

$f (- 5) = \frac{2 (11)}{4}$

Step 3.5.3.3.2

Cancel the common factors.

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Step 3.5.3.3.2.1

Factor $2$ out of $4$ .

$f (- 5) = \frac{2 \cdot 11}{2 \cdot 2}$

Step 3.5.3.3.2.2

Cancel the common factor.

$f (- 5) = \frac{2 \cdot 11}{2 \cdot 2}$

Step 3.5.3.3.2.3

Rewrite the expression.

$f (- 5) = \frac{11}{2}$

Step 3.5.4

The final answer is $\frac{11}{2}$ .

$\frac{11}{2}$

Step 3.6

The $y$ value at $x = - 5$ is $\frac{11}{2}$ .

$y = \frac{11}{2}$

Step 3.7

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

$f (- 2) = \frac{{(- 2)}^{2}}{8} + \frac{3 (- 2)}{4} + \frac{49}{8}$

Step 3.8

Simplify the result.

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Step 3.8.1

Combine fractions.

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Step 3.8.1.1

Combine the numerators over the common denominator.

$f (- 2) = \frac{{(- 2)}^{2} + 49}{8} + \frac{3 (- 2)}{4}$

Step 3.8.1.2

Simplify the expression.

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Step 3.8.1.2.1

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

$f (- 2) = \frac{4 + 49}{8} + \frac{3 (- 2)}{4}$

Step 3.8.1.2.2

Add $4$ and $49$ .

$f (- 2) = \frac{53}{8} + \frac{3 (- 2)}{4}$

Step 3.8.1.2.3

Multiply $3$ by $- 2$ .

$f (- 2) = \frac{53}{8} + \frac{- 6}{4}$

Step 3.8.2

Simplify each term.

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Step 3.8.2.1

Cancel the common factor of $- 6$ and $4$ .

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Step 3.8.2.1.1

Factor $2$ out of $- 6$ .

$f (- 2) = \frac{53}{8} + \frac{2 (- 3)}{4}$

Step 3.8.2.1.2

Cancel the common factors.

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Step 3.8.2.1.2.1

Factor $2$ out of $4$ .

$f (- 2) = \frac{53}{8} + \frac{2 \cdot - 3}{2 \cdot 2}$

Step 3.8.2.1.2.2

Cancel the common factor.

$f (- 2) = \frac{53}{8} + \frac{2 \cdot - 3}{2 \cdot 2}$

Step 3.8.2.1.2.3

Rewrite the expression.

$f (- 2) = \frac{53}{8} + \frac{- 3}{2}$

Step 3.8.2.2

Move the negative in front of the fraction.

$f (- 2) = \frac{53}{8} - \frac{3}{2}$

Step 3.8.3

To write $- \frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (- 2) = \frac{53}{8} - \frac{3}{2} \cdot \frac{4}{4}$

Step 3.8.4

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.8.4.1

Multiply $\frac{3}{2}$ by $\frac{4}{4}$ .

$f (- 2) = \frac{53}{8} - \frac{3 \cdot 4}{2 \cdot 4}$

Step 3.8.4.2

Multiply $2$ by $4$ .

$f (- 2) = \frac{53}{8} - \frac{3 \cdot 4}{8}$

Step 3.8.5

Combine the numerators over the common denominator.

$f (- 2) = \frac{53 - 3 \cdot 4}{8}$

Step 3.8.6

Simplify the numerator.

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Step 3.8.6.1

Multiply $- 3$ by $4$ .

$f (- 2) = \frac{53 - 12}{8}$

Step 3.8.6.2

Subtract $12$ from $53$ .

$f (- 2) = \frac{41}{8}$

Step 3.8.7

The final answer is $\frac{41}{8}$ .

$\frac{41}{8}$

Step 3.9

The $y$ value at $x = - 2$ is $\frac{41}{8}$ .

$y = \frac{41}{8}$

Step 3.10

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

$f (- 1) = \frac{{(- 1)}^{2}}{8} + \frac{3 (- 1)}{4} + \frac{49}{8}$

Step 3.11

Simplify the result.

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Step 3.11.1

Combine fractions.

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Step 3.11.1.1

Combine the numerators over the common denominator.

$f (- 1) = \frac{{(- 1)}^{2} + 49}{8} + \frac{3 (- 1)}{4}$

Step 3.11.1.2

Simplify the expression.

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Step 3.11.1.2.1

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

$f (- 1) = \frac{1 + 49}{8} + \frac{3 (- 1)}{4}$

Step 3.11.1.2.2

Add $1$ and $49$ .

$f (- 1) = \frac{50}{8} + \frac{3 (- 1)}{4}$

Step 3.11.1.2.3

Multiply $3$ by $- 1$ .

$f (- 1) = \frac{50}{8} + \frac{- 3}{4}$

Step 3.11.2

Simplify each term.

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Step 3.11.2.1

Cancel the common factor of $50$ and $8$ .

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Step 3.11.2.1.1

Factor $2$ out of $50$ .

$f (- 1) = \frac{2 (25)}{8} + \frac{- 3}{4}$

Step 3.11.2.1.2

Cancel the common factors.

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Step 3.11.2.1.2.1

Factor $2$ out of $8$ .

$f (- 1) = \frac{2 \cdot 25}{2 \cdot 4} + \frac{- 3}{4}$

Step 3.11.2.1.2.2

Cancel the common factor.

$f (- 1) = \frac{2 \cdot 25}{2 \cdot 4} + \frac{- 3}{4}$

Step 3.11.2.1.2.3

Rewrite the expression.

$f (- 1) = \frac{25}{4} + \frac{- 3}{4}$

Step 3.11.2.2

Move the negative in front of the fraction.

$f (- 1) = \frac{25}{4} - \frac{3}{4}$

Step 3.11.3

Simplify terms.

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Step 3.11.3.1

Combine the numerators over the common denominator.

$f (- 1) = \frac{25 - 3}{4}$

Step 3.11.3.2

Subtract $3$ from $25$ .

$f (- 1) = \frac{22}{4}$

Step 3.11.3.3

Cancel the common factor of $22$ and $4$ .

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Step 3.11.3.3.1

Factor $2$ out of $22$ .

$f (- 1) = \frac{2 (11)}{4}$

Step 3.11.3.3.2

Cancel the common factors.

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Step 3.11.3.3.2.1

Factor $2$ out of $4$ .

$f (- 1) = \frac{2 \cdot 11}{2 \cdot 2}$

Step 3.11.3.3.2.2

Cancel the common factor.

$f (- 1) = \frac{2 \cdot 11}{2 \cdot 2}$

Step 3.11.3.3.2.3

Rewrite the expression.

$f (- 1) = \frac{11}{2}$

Step 3.11.4

The final answer is $\frac{11}{2}$ .

$\frac{11}{2}$

Step 3.12

The $y$ value at $x = - 1$ is $\frac{11}{2}$ .

$y = \frac{11}{2}$

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 5 & \frac{11}{2} \\ - 4 & \frac{41}{8} \\ - 3 & 5 \\ - 2 & \frac{41}{8} \\ - 1 & \frac{11}{2} \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 3, 5)$

Focus: $(- 3, 7)$

Axis of Symmetry: $x = - 3$

Directrix: $y = 3$

$\begin{array}{c} x & y \\ - 5 & \frac{11}{2} \\ - 4 & \frac{41}{8} \\ - 3 & 5 \\ - 2 & \frac{41}{8} \\ - 1 & \frac{11}{2} \end{array}$

Step 5