Trigonometry Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 16$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Divide $x$ by $1$ .

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

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

Step 3

Find the properties of the given parabola.

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Step 3.1

Rewrite the equation in vertex form.

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Step 3.1.1

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

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Step 3.1.1.1

Negate $\frac{{(y - 2)}^{2}}{16}$ .

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

Step 3.1.1.2

Reorder terms.

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

Step 3.1.2

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

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Step 3.1.2.1

Simplify the expression.

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Step 3.1.2.1.1

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

$- \frac{1}{16} \cdot ((y - 2) (y - 2))$

Step 3.1.2.1.2

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

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Step 3.1.2.1.2.1

Apply the distributive property.

$- \frac{1}{16} \cdot (y (y - 2) - 2 (y - 2))$

Step 3.1.2.1.2.2

Apply the distributive property.

$- \frac{1}{16} \cdot (y \cdot y + y \cdot - 2 - 2 (y - 2))$

Step 3.1.2.1.2.3

Apply the distributive property.

$- \frac{1}{16} \cdot (y \cdot y + y \cdot - 2 - 2 y - 2 \cdot - 2)$

Step 3.1.2.1.3

Simplify and combine like terms.

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Step 3.1.2.1.3.1

Simplify each term.

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Step 3.1.2.1.3.1.1

Multiply $y$ by $y$ .

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

Step 3.1.2.1.3.1.2

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

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

Step 3.1.2.1.3.1.3

Multiply $- 2$ by $- 2$ .

$- \frac{1}{16} \cdot (y^{2} - 2 y - 2 y + 4)$

Step 3.1.2.1.3.2

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

$- \frac{1}{16} \cdot (y^{2} - 4 y + 4)$

Step 3.1.2.1.4

Apply the distributive property.

$- \frac{1}{16} y^{2} - \frac{1}{16} (- 4 y) - \frac{1}{16} \cdot 4$

Step 3.1.2.1.5

Simplify.

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Step 3.1.2.1.5.1

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

$- \frac{y^{2}}{16} - \frac{1}{16} (- 4 y) - \frac{1}{16} \cdot 4$

Step 3.1.2.1.5.2

Cancel the common factor of $4$ .

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Step 3.1.2.1.5.2.1

Move the leading negative in $- \frac{1}{16}$ into the numerator.

$- \frac{y^{2}}{16} + \frac{- 1}{16} (- 4 y) - \frac{1}{16} \cdot 4$

Step 3.1.2.1.5.2.2

Factor $4$ out of $16$ .

$- \frac{y^{2}}{16} + \frac{- 1}{4 (4)} (- 4 y) - \frac{1}{16} \cdot 4$

Step 3.1.2.1.5.2.3

Factor $4$ out of $- 4 y$ .

$- \frac{y^{2}}{16} + \frac{- 1}{4 (4)} (4 (- y)) - \frac{1}{16} \cdot 4$

Step 3.1.2.1.5.2.4

Cancel the common factor.

$- \frac{y^{2}}{16} + \frac{- 1}{4 \cdot 4} (4 (- y)) - \frac{1}{16} \cdot 4$

Step 3.1.2.1.5.2.5

Rewrite the expression.

$- \frac{y^{2}}{16} + \frac{- 1}{4} (- y) - \frac{1}{16} \cdot 4$

Step 3.1.2.1.5.3

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

$- \frac{y^{2}}{16} - \frac{- y}{4} - \frac{1}{16} \cdot 4$

Step 3.1.2.1.5.4

Cancel the common factor of $4$ .

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Step 3.1.2.1.5.4.1

Move the leading negative in $- \frac{1}{16}$ into the numerator.

$- \frac{y^{2}}{16} - \frac{- y}{4} + \frac{- 1}{16} \cdot 4$

Step 3.1.2.1.5.4.2

Factor $4$ out of $16$ .

$- \frac{y^{2}}{16} - \frac{- y}{4} + \frac{- 1}{4 (4)} \cdot 4$

Step 3.1.2.1.5.4.3

Cancel the common factor.

$- \frac{y^{2}}{16} - \frac{- y}{4} + \frac{- 1}{4 \cdot 4} \cdot 4$

Step 3.1.2.1.5.4.4

Rewrite the expression.

$- \frac{y^{2}}{16} - \frac{- y}{4} + \frac{- 1}{4}$

Step 3.1.2.1.6

Simplify each term.

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Step 3.1.2.1.6.1

Move the negative in front of the fraction.

$- \frac{y^{2}}{16} - - \frac{y}{4} + \frac{- 1}{4}$

Step 3.1.2.1.6.2

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

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Step 3.1.2.1.6.2.1

Multiply $- 1$ by $- 1$ .

$- \frac{y^{2}}{16} + 1 \frac{y}{4} + \frac{- 1}{4}$

Step 3.1.2.1.6.2.2

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

$- \frac{y^{2}}{16} + \frac{y}{4} + \frac{- 1}{4}$

Step 3.1.2.1.6.3

Move the negative in front of the fraction.

$- \frac{y^{2}}{16} + \frac{y}{4} - \frac{1}{4}$

Step 3.1.2.2

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

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

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

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

Step 3.1.2.3

Consider the vertex form of a parabola.

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

Step 3.1.2.4

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

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Step 3.1.2.4.1

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

$d = \frac{\frac{1}{4}}{2 (- \frac{1}{16})}$

Step 3.1.2.4.2

Simplify the right side.

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Step 3.1.2.4.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{1}{4} \cdot \frac{1}{2 (- \frac{1}{16})}$

Step 3.1.2.4.2.2

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

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Step 3.1.2.4.2.2.1

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

$d = \frac{1}{4} \cdot \frac{- 1 (- 1)}{2 (- \frac{1}{16})}$

Step 3.1.2.4.2.2.2

Move the negative in front of the fraction.

$d = \frac{1}{4} (- \frac{1}{2 (\frac{1}{16})})$

Step 3.1.2.4.2.3

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

$d = \frac{1}{4} (- \frac{1}{\frac{2}{16}})$

Step 3.1.2.4.2.4

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

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Step 3.1.2.4.2.4.1

Factor $2$ out of $2$ .

$d = \frac{1}{4} (- \frac{1}{\frac{2 (1)}{16}})$

Step 3.1.2.4.2.4.2

Cancel the common factors.

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Step 3.1.2.4.2.4.2.1

Factor $2$ out of $16$ .

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

Step 3.1.2.4.2.4.2.2

Cancel the common factor.

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

Step 3.1.2.4.2.4.2.3

Rewrite the expression.

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

Step 3.1.2.4.2.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.1.2.4.2.6

Multiply $- (1 \cdot 8)$ .

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Step 3.1.2.4.2.6.1

Multiply $8$ by $1$ .

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

Step 3.1.2.4.2.6.2

Multiply $- 1$ by $8$ .

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

Step 3.1.2.4.2.7

Cancel the common factor of $4$ .

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Step 3.1.2.4.2.7.1

Factor $4$ out of $- 8$ .

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

Step 3.1.2.4.2.7.2

Cancel the common factor.

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

Step 3.1.2.4.2.7.3

Rewrite the expression.

$d = - 2$

Step 3.1.2.5

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

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Step 3.1.2.5.1

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

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

Step 3.1.2.5.2

Simplify the right side.

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Step 3.1.2.5.2.1

Simplify each term.

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Step 3.1.2.5.2.1.1

Simplify the numerator.

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Step 3.1.2.5.2.1.1.1

Apply the product rule to $\frac{1}{4}$ .

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

Step 3.1.2.5.2.1.1.2

One to any power is one.

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

Step 3.1.2.5.2.1.1.3

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

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

Step 3.1.2.5.2.1.2

Simplify the denominator.

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Step 3.1.2.5.2.1.2.1

Multiply $- 1$ by $4$ .

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

Step 3.1.2.5.2.1.2.2

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

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

Step 3.1.2.5.2.1.3

Reduce the expression by cancelling the common factors.

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Step 3.1.2.5.2.1.3.1

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

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Step 3.1.2.5.2.1.3.1.1

Factor $4$ out of $- 4$ .

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

Step 3.1.2.5.2.1.3.1.2

Cancel the common factors.

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Step 3.1.2.5.2.1.3.1.2.1

Factor $4$ out of $16$ .

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

Step 3.1.2.5.2.1.3.1.2.2

Cancel the common factor.

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

Step 3.1.2.5.2.1.3.1.2.3

Rewrite the expression.

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

Step 3.1.2.5.2.1.3.2

Move the negative in front of the fraction.

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

Step 3.1.2.5.2.1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.1.2.5.2.1.5

Cancel the common factor of $4$ .

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Step 3.1.2.5.2.1.5.1

Factor $4$ out of $16$ .

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

Step 3.1.2.5.2.1.5.2

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

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

Step 3.1.2.5.2.1.5.3

Cancel the common factor.

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

Step 3.1.2.5.2.1.5.4

Rewrite the expression.

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

Step 3.1.2.5.2.1.6

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

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

Step 3.1.2.5.2.1.7

Move the negative in front of the fraction.

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

Step 3.1.2.5.2.1.8

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

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Step 3.1.2.5.2.1.8.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.2.5.2.1.8.2

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

$e = - \frac{1}{4} + \frac{1}{4}$

Step 3.1.2.5.2.2

Combine the numerators over the common denominator.

$e = \frac{- 1 + 1}{4}$

Step 3.1.2.5.2.3

Add $- 1$ and $1$ .

$e = \frac{0}{4}$

Step 3.1.2.5.2.4

Divide $0$ by $4$ .

$e = 0$

Step 3.1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{16} {(y - 2)}^{2} + 0$ .

$- \frac{1}{16} {(y - 2)}^{2} + 0$

Step 3.1.3

Set $x$ equal to the new right side.

$x = - \frac{1}{16} \cdot {(y - 2)}^{2} + 0$

Step 3.2

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

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

$h = 0$

$k = 2$

Step 3.3

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

Opens Left

Step 3.4

Find the vertex $(h, k)$ .

$(0, 2)$

Step 3.5

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

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Step 3.5.1

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

$\frac{1}{4 a}$

Step 3.5.2

Substitute the value of $a$ into the formula.

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

Step 3.5.3

Simplify.

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Step 3.5.3.1

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

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Step 3.5.3.1.1

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

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

Step 3.5.3.1.2

Move the negative in front of the fraction.

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

Step 3.5.3.2

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

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

Step 3.5.3.3

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

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Step 3.5.3.3.1

Factor $4$ out of $4$ .

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

Step 3.5.3.3.2

Cancel the common factors.

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Step 3.5.3.3.2.1

Factor $4$ out of $16$ .

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

Step 3.5.3.3.2.2

Cancel the common factor.

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

Step 3.5.3.3.2.3

Rewrite the expression.

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

Step 3.5.3.4

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 4)$

Step 3.5.3.5

Multiply $- (1 \cdot 4)$ .

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Step 3.5.3.5.1

Multiply $4$ by $1$ .

$- 1 \cdot 4$

Step 3.5.3.5.2

Multiply $- 1$ by $4$ .

$- 4$

Step 3.6

Find the focus.

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Step 3.6.1

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 3.6.2

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

$(- 4, 2)$

Step 3.7

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

$y = 2$

Step 3.8

Find the directrix.

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Step 3.8.1

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 3.8.2

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

$x = 4$

Step 3.9

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

Direction: Opens Left

Vertex: $(0, 2)$

Focus: $(- 4, 2)$

Axis of Symmetry: $y = 2$

Directrix: $x = 4$

Direction: Opens Left

Vertex: $(0, 2)$

Focus: $(- 4, 2)$

Axis of Symmetry: $y = 2$

Directrix: $x = 4$

Step 4

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 4.1

Substitute the $x$ value $- 2$ into $f (x) = 4 \sqrt{- 1 x} + 2$ . In this case, the point is $(- 2, 7.65685424)$ .

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Step 4.1.1

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

$f (- 2) = 4 \sqrt{- 1 \cdot - 2} + 2$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Multiply $- 1$ by $- 2$ .

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

Step 4.1.2.2

The final answer is $4 \sqrt{2} + 2$ .

$y = 4 \sqrt{2} + 2$

Step 4.1.3

Convert $4 \sqrt{2} + 2$ to decimal.

$= 7.65685424$

Step 4.2

Substitute the $x$ value $- 2$ into $f (x) = - 4 \sqrt{- 1 x} + 2$ . In this case, the point is $(- 2, - 3.65685424)$ .

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Step 4.2.1

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

$f (- 2) = - 4 \sqrt{- 1 \cdot - 2} + 2$

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Multiply $- 1$ by $- 2$ .

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

Step 4.2.2.2

The final answer is $- 4 \sqrt{2} + 2$ .

$y = - 4 \sqrt{2} + 2$

Step 4.2.3

Convert $- 4 \sqrt{2} + 2$ to decimal.

$= - 3.65685424$

Step 4.3

Substitute the $x$ value $- 1$ into $f (x) = 4 \sqrt{- 1 x} + 2$ . In this case, the point is $(- 1, 6)$ .

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Step 4.3.1

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

$f (- 1) = 4 \sqrt{- 1 \cdot - 1} + 2$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Multiply $- 1$ by $- 1$ .

$f (- 1) = 4 \sqrt{1} + 2$

Step 4.3.2.1.2

Any root of $1$ is $1$ .

$f (- 1) = 4 \cdot 1 + 2$

Step 4.3.2.1.3

Multiply $4$ by $1$ .

$f (- 1) = 4 + 2$

Step 4.3.2.2

Add $4$ and $2$ .

$f (- 1) = 6$

Step 4.3.2.3

The final answer is $6$ .

$y = 6$

Step 4.3.3

Convert $6$ to decimal.

$= 6$

Step 4.4

Substitute the $x$ value $- 1$ into $f (x) = - 4 \sqrt{- 1 x} + 2$ . In this case, the point is $(- 1, - 2)$ .

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Step 4.4.1

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

$f (- 1) = - 4 \sqrt{- 1 \cdot - 1} + 2$

Step 4.4.2

Simplify the result.

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Step 4.4.2.1

Simplify each term.

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Step 4.4.2.1.1

Multiply $- 1$ by $- 1$ .

$f (- 1) = - 4 \sqrt{1} + 2$

Step 4.4.2.1.2

Any root of $1$ is $1$ .

$f (- 1) = - 4 \cdot 1 + 2$

Step 4.4.2.1.3

Multiply $- 4$ by $1$ .

$f (- 1) = - 4 + 2$

Step 4.4.2.2

Add $- 4$ and $2$ .

$f (- 1) = - 2$

Step 4.4.2.3

The final answer is $- 2$ .

$y = - 2$

Step 4.4.3

Convert $- 2$ to decimal.

$= - 2$

Step 4.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & 7.66 \\ - 2 & - 3.66 \\ - 1 & 6 \\ - 1 & - 2 \\ 0 & 2 \end{array}$

Step 5

Graph the parabola using its properties and the selected points.

Direction: Opens Left

Vertex: $(0, 2)$

Focus: $(- 4, 2)$

Axis of Symmetry: $y = 2$

Directrix: $x = 4$

$\begin{array}{c} x & y \\ - 2 & 7.66 \\ - 2 & - 3.66 \\ - 1 & 6 \\ - 1 & - 2 \\ 0 & 2 \end{array}$

Step 6