Pre-Algebra Examples

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

Step 1

Simplify $- \frac{1}{5} \cdot {(y - 2)}^{2} - 4$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite ${(y - 2)}^{2}$ as $(y - 2) (y - 2)$ .

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

Step 1.1.2

Expand $(y - 2) (y - 2)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

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

Step 1.1.2.2

Apply the distributive property.

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

Step 1.1.2.3

Apply the distributive property.

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

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply $y$ by $y$ .

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

Step 1.1.3.1.2

Move $- 2$ to the left of $y$ .

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

Step 1.1.3.1.3

Multiply $- 2$ by $- 2$ .

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

Step 1.1.3.2

Subtract $2 y$ from $- 2 y$ .

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

Step 1.1.4

Apply the distributive property.

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

Step 1.1.5

Simplify.

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Step 1.1.5.1

Combine $y^{2}$ and $\frac{1}{5}$ .

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

Step 1.1.5.2

Multiply $- \frac{1}{5} (- 4 y)$ .

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Step 1.1.5.2.1

Multiply $- 4$ by $- 1$ .

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

Step 1.1.5.2.2

Combine $4$ and $\frac{1}{5}$ .

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

Step 1.1.5.2.3

Combine $\frac{4}{5}$ and $y$ .

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

Step 1.1.5.3

Multiply $- \frac{1}{5} \cdot 4$ .

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Step 1.1.5.3.1

Multiply $4$ by $- 1$ .

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

Step 1.1.5.3.2

Combine $- 4$ and $\frac{1}{5}$ .

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

Step 1.1.6

Move the negative in front of the fraction.

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

Step 1.2

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x = - \frac{y^{2}}{5} + \frac{4 y}{5} - \frac{4}{5} - 4 \cdot \frac{5}{5}$

Step 1.3

Combine $- 4$ and $\frac{5}{5}$ .

$x = - \frac{y^{2}}{5} + \frac{4 y}{5} - \frac{4}{5} + \frac{- 4 \cdot 5}{5}$

Step 1.4

Combine the numerators over the common denominator.

$x = - \frac{y^{2}}{5} + \frac{4 y}{5} + \frac{- 4 - 4 \cdot 5}{5}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $- 4$ by $5$ .

$x = - \frac{y^{2}}{5} + \frac{4 y}{5} + \frac{- 4 - 20}{5}$

Step 1.5.2

Subtract $20$ from $- 4$ .

$x = - \frac{y^{2}}{5} + \frac{4 y}{5} + \frac{- 24}{5}$

Step 1.6

Move the negative in front of the fraction.

$x = - \frac{y^{2}}{5} + \frac{4 y}{5} - \frac{24}{5}$

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 $- \frac{y^{2}}{5} + \frac{4 y}{5} - \frac{24}{5}$ .

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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 $c$ .

$a = - \frac{1}{5}$

$b = \frac{4}{5}$

$c = - \frac{24}{5}$

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

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

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.

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

Step 2.1.1.3.2.2

Cancel the common factor of $2$ .

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Step 2.1.1.3.2.2.1

Factor $2$ out of $4$ .

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

Step 2.1.1.3.2.2.2

Cancel the common factor.

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

Step 2.1.1.3.2.2.3

Rewrite the expression.

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

Step 2.1.1.3.2.3

Multiply $\frac{2}{5}$ by $\frac{1}{- \frac{1}{5}}$ .

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

Step 2.1.1.3.2.4

Multiply $- 1$ by $5$ .

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

Step 2.1.1.3.2.5

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

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

Step 2.1.1.3.2.6

Cancel the common factor of $- 5$ and $5$ .

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Step 2.1.1.3.2.6.1

Factor $5$ out of $- 5$ .

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

Step 2.1.1.3.2.6.2

Cancel the common factors.

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Step 2.1.1.3.2.6.2.1

Factor $5$ out of $5$ .

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

Step 2.1.1.3.2.6.2.2

Cancel the common factor.

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

Step 2.1.1.3.2.6.2.3

Rewrite the expression.

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

Step 2.1.1.3.2.6.2.4

Divide $- 1$ by $1$ .

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

Step 2.1.1.3.2.7

Move the negative one from the denominator of $\frac{2}{- 1}$ .

$d = - 1 \cdot 2$

Step 2.1.1.3.2.8

Multiply $- 1$ by $2$ .

$d = - 2$

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

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

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 $\frac{4}{5}$ .

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

Step 2.1.1.4.2.1.1.2

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

$e = - \frac{24}{5} - \frac{\frac{16}{5^{2}}}{4 (- \frac{1}{5})}$

Step 2.1.1.4.2.1.1.3

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

$e = - \frac{24}{5} - \frac{\frac{16}{25}}{4 (- \frac{1}{5})}$

Step 2.1.1.4.2.1.2

Simplify the denominator.

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Step 2.1.1.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = - \frac{24}{5} - \frac{\frac{16}{25}}{- 4 (\frac{1}{5})}$

Step 2.1.1.4.2.1.2.2

Combine $- 4$ and $\frac{1}{5}$ .

$e = - \frac{24}{5} - \frac{\frac{16}{25}}{\frac{- 4}{5}}$

Step 2.1.1.4.2.1.3

Move the negative in front of the fraction.

$e = - \frac{24}{5} - \frac{\frac{16}{25}}{- \frac{4}{5}}$

Step 2.1.1.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{24}{5} - (\frac{16}{25} (- \frac{5}{4}))$

Step 2.1.1.4.2.1.5

Cancel the common factor of $4$ .

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Step 2.1.1.4.2.1.5.1

Move the leading negative in $- \frac{5}{4}$ into the numerator.

$e = - \frac{24}{5} - (\frac{16}{25} \cdot \frac{- 5}{4})$

Step 2.1.1.4.2.1.5.2

Factor $4$ out of $16$ .

$e = - \frac{24}{5} - (\frac{4 (4)}{25} \cdot \frac{- 5}{4})$

Step 2.1.1.4.2.1.5.3

Cancel the common factor.

$e = - \frac{24}{5} - (\frac{4 \cdot 4}{25} \cdot \frac{- 5}{4})$

Step 2.1.1.4.2.1.5.4

Rewrite the expression.

$e = - \frac{24}{5} - (\frac{4}{25} \cdot - 5)$

Step 2.1.1.4.2.1.6

Cancel the common factor of $5$ .

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Step 2.1.1.4.2.1.6.1

Factor $5$ out of $25$ .

$e = - \frac{24}{5} - (\frac{4}{5 (5)} \cdot - 5)$

Step 2.1.1.4.2.1.6.2

Factor $5$ out of $- 5$ .

$e = - \frac{24}{5} - (\frac{4}{5 \cdot 5} \cdot (5 \cdot - 1))$

Step 2.1.1.4.2.1.6.3

Cancel the common factor.

$e = - \frac{24}{5} - (\frac{4}{5 \cdot 5} \cdot (5 \cdot - 1))$

Step 2.1.1.4.2.1.6.4

Rewrite the expression.

$e = - \frac{24}{5} - (\frac{4}{5} \cdot - 1)$

Step 2.1.1.4.2.1.7

Combine $\frac{4}{5}$ and $- 1$ .

$e = - \frac{24}{5} - \frac{4 \cdot - 1}{5}$

Step 2.1.1.4.2.1.8

Multiply $4$ by $- 1$ .

$e = - \frac{24}{5} - \frac{- 4}{5}$

Step 2.1.1.4.2.1.9

Move the negative in front of the fraction.

$e = - \frac{24}{5} - - \frac{4}{5}$

Step 2.1.1.4.2.1.10

Multiply $- - \frac{4}{5}$ .

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Step 2.1.1.4.2.1.10.1

Multiply $- 1$ by $- 1$ .

$e = - \frac{24}{5} + 1 (\frac{4}{5})$

Step 2.1.1.4.2.1.10.2

Multiply $\frac{4}{5}$ by $1$ .

$e = - \frac{24}{5} + \frac{4}{5}$

Step 2.1.1.4.2.2

Combine the numerators over the common denominator.

$e = \frac{- 24 + 4}{5}$

Step 2.1.1.4.2.3

Add $- 24$ and $4$ .

$e = \frac{- 20}{5}$

Step 2.1.1.4.2.4

Divide $- 20$ by $5$ .

$e = - 4$

Step 2.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{5} {(y - 2)}^{2} - 4$ .

$- \frac{1}{5} {(y - 2)}^{2} - 4$

Step 2.1.2

Set $x$ equal to the new right side.

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

Step 2.2

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

$a = - \frac{1}{5}$

$h = - 4$

$k = 2$

Step 2.3

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

Opens Left

Step 2.4

Find the vertex $(h, k)$ .

$(- 4, 2)$

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.

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

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 $- 1 (- 1)$ .

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

Step 2.5.3.1.2

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{5})}$

Step 2.5.3.2

Combine $4$ and $\frac{1}{5}$ .

$- \frac{1}{\frac{4}{5}}$

Step 2.5.3.3

Multiply the numerator by the reciprocal of the denominator.

$- (1 (\frac{5}{4}))$

Step 2.5.3.4

Multiply $\frac{5}{4}$ by $1$ .

$- \frac{5}{4}$

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 x-coordinate $h$ if the parabola opens left or right.

$(h + p, k)$

Step 2.6.2

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

$(- \frac{21}{4}, 2)$

Step 2.7

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

$y = 2$

Step 2.8

Find the directrix.

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Step 2.8.1

The directrix of a parabola is the vertical line found by subtracting $p$ from the x-coordinate $h$ of the vertex if the parabola opens left or right.

$x = h - p$

Step 2.8.2

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

$x = - \frac{11}{4}$

Step 2.9

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

Direction: Opens Left

Vertex: $(- 4, 2)$

Focus: $(- \frac{21}{4}, 2)$

Axis of Symmetry: $y = 2$

Directrix: $x = - \frac{11}{4}$

Direction: Opens Left

Vertex: $(- 4, 2)$

Focus: $(- \frac{21}{4}, 2)$

Axis of Symmetry: $y = 2$

Directrix: $x = - \frac{11}{4}$

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

Substitute the $x$ value $- 6$ into $f (x) = \sqrt{5 (- x - 4)} + 2$ . In this case, the point is $(- 6, 5.16227766)$ .

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Step 3.1.1

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

$f (- 6) = \sqrt{5 (- (- 6) - 4)} + 2$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Multiply $- 1$ by $- 6$ .

$f (- 6) = \sqrt{5 (6 - 4)} + 2$

Step 3.1.2.1.2

Subtract $4$ from $6$ .

$f (- 6) = \sqrt{5 \cdot 2} + 2$

Step 3.1.2.1.3

Multiply $5$ by $2$ .

$f (- 6) = \sqrt{10} + 2$

Step 3.1.2.2

The final answer is $\sqrt{10} + 2$ .

$y = \sqrt{10} + 2$

Step 3.1.3

Convert $\sqrt{10} + 2$ to decimal.

$= 5.16227766$

Step 3.2

Substitute the $x$ value $- 6$ into $f (x) = - \sqrt{5 (- x - 4)} + 2$ . In this case, the point is $(- 6, - 1.16227766)$ .

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Step 3.2.1

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

$f (- 6) = - \sqrt{5 (- (- 6) - 4)} + 2$

Step 3.2.2

Simplify the result.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Multiply $- 1$ by $- 6$ .

$f (- 6) = - \sqrt{5 (6 - 4)} + 2$

Step 3.2.2.1.2

Subtract $4$ from $6$ .

$f (- 6) = - \sqrt{5 \cdot 2} + 2$

Step 3.2.2.1.3

Multiply $5$ by $2$ .

$f (- 6) = - \sqrt{10} + 2$

Step 3.2.2.2

The final answer is $- \sqrt{10} + 2$ .

$y = - \sqrt{10} + 2$

Step 3.2.3

Convert $- \sqrt{10} + 2$ to decimal.

$= - 1.16227766$

Step 3.3

Substitute the $x$ value $- 5$ into $f (x) = \sqrt{5 (- x - 4)} + 2$ . In this case, the point is $(- 5, 4.23606797)$ .

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Step 3.3.1

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

$f (- 5) = \sqrt{5 (- (- 5) - 4)} + 2$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

Multiply $- 1$ by $- 5$ .

$f (- 5) = \sqrt{5 (5 - 4)} + 2$

Step 3.3.2.1.2

Subtract $4$ from $5$ .

$f (- 5) = \sqrt{5 \cdot 1} + 2$

Step 3.3.2.1.3

Multiply $5$ by $1$ .

$f (- 5) = \sqrt{5} + 2$

Step 3.3.2.2

The final answer is $\sqrt{5} + 2$ .

$y = \sqrt{5} + 2$

Step 3.3.3

Convert $\sqrt{5} + 2$ to decimal.

$= 4.23606797$

Step 3.4

Substitute the $x$ value $- 5$ into $f (x) = - \sqrt{5 (- x - 4)} + 2$ . In this case, the point is $(- 5, - 0.23606797)$ .

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Step 3.4.1

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

$f (- 5) = - \sqrt{5 (- (- 5) - 4)} + 2$

Step 3.4.2

Simplify the result.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

Multiply $- 1$ by $- 5$ .

$f (- 5) = - \sqrt{5 (5 - 4)} + 2$

Step 3.4.2.1.2

Subtract $4$ from $5$ .

$f (- 5) = - \sqrt{5 \cdot 1} + 2$

Step 3.4.2.1.3

Multiply $5$ by $1$ .

$f (- 5) = - \sqrt{5} + 2$

Step 3.4.2.2

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

$y = - \sqrt{5} + 2$

Step 3.4.3

Convert $- \sqrt{5} + 2$ to decimal.

$= - 0.23606797$

Step 3.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 6 & 5.16 \\ - 6 & - 1.16 \\ - 5 & 4.24 \\ - 5 & - 0.24 \\ - 4 & 2 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Left

Vertex: $(- 4, 2)$

Focus: $(- \frac{21}{4}, 2)$

Axis of Symmetry: $y = 2$

Directrix: $x = - \frac{11}{4}$

$\begin{array}{c} x & y \\ - 6 & 5.16 \\ - 6 & - 1.16 \\ - 5 & 4.24 \\ - 5 & - 0.24 \\ - 4 & 2 \end{array}$

Step 5