Precalculus Examples

$y^{2} - 4 y + 4 x + 4 = 0$

Step 1

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

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Step 1.1

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

$- 4 y + 4 x + 4 = - y^{2}$

Step 1.2

Add $4 y$ to both sides of the equation.

$4 x + 4 = - y^{2} + 4 y$

Step 1.3

Subtract $4$ from both sides of the equation.

$4 x = - y^{2} + 4 y - 4$

Step 2

Divide each term in $4 x = - y^{2} + 4 y - 4$ by $4$ and simplify.

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Step 2.1

Divide each term in $4 x = - y^{2} + 4 y - 4$ by $4$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Divide $x$ by $1$ .

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Move the negative in front of the fraction.

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

Step 2.3.1.2

Cancel the common factor of $4$ .

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Step 2.3.1.2.1

Cancel the common factor.

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

Step 2.3.1.2.2

Divide $y$ by $1$ .

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

Step 2.3.1.3

Divide $- 4$ by $4$ .

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

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

Complete the square for $- \frac{y^{2}}{4} + y - 1$ .

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Step 3.1.1.1

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

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

$b = 1$

$c = - 1$

Step 3.1.1.2

Consider the vertex form of a parabola.

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

Step 3.1.1.3

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

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Step 3.1.1.3.1

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

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

Step 3.1.1.3.2

Simplify the right side.

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Step 3.1.1.3.2.1

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

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Step 3.1.1.3.2.1.1

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

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

Step 3.1.1.3.2.1.2

Move the negative in front of the fraction.

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

Step 3.1.1.3.2.2

Combine $2$ and $\frac{1}{4}$ .

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

Step 3.1.1.3.2.3

Cancel the common factor of $2$ and $4$ .

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Step 3.1.1.3.2.3.1

Factor $2$ out of $2$ .

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

Step 3.1.1.3.2.3.2

Cancel the common factors.

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Step 3.1.1.3.2.3.2.1

Factor $2$ out of $4$ .

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

Step 3.1.1.3.2.3.2.2

Cancel the common factor.

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

Step 3.1.1.3.2.3.2.3

Rewrite the expression.

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

Step 3.1.1.3.2.4

Multiply the numerator by the reciprocal of the denominator.

$d = - (1 \cdot 2)$

Step 3.1.1.3.2.5

Multiply $- (1 \cdot 2)$ .

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Step 3.1.1.3.2.5.1

Multiply $2$ by $1$ .

$d = - 1 \cdot 2$

Step 3.1.1.3.2.5.2

Multiply $- 1$ by $2$ .

$d = - 2$

Step 3.1.1.4

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

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Step 3.1.1.4.1

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

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

Step 3.1.1.4.2

Simplify the right side.

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Step 3.1.1.4.2.1

Simplify each term.

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Step 3.1.1.4.2.1.1

One to any power is one.

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

Step 3.1.1.4.2.1.2

Simplify the denominator.

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Step 3.1.1.4.2.1.2.1

Multiply $- 1$ by $4$ .

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

Step 3.1.1.4.2.1.2.2

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

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

Step 3.1.1.4.2.1.3

Divide $- 4$ by $4$ .

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

Step 3.1.1.4.2.1.4

Divide $1$ by $- 1$ .

$e = - 1 - - 1$

Step 3.1.1.4.2.1.5

Multiply $- 1$ by $- 1$ .

$e = - 1 + 1$

Step 3.1.1.4.2.2

Add $- 1$ and $1$ .

$e = 0$

Step 3.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{4} {(y - 2)}^{2} + 0$ .

$- \frac{1}{4} {(y - 2)}^{2} + 0$

Step 3.1.2

Set $x$ equal to the new right side.

$x = - \frac{1}{4} \cdot {(y - 2)}^{2} + 0$

Step 3.2

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

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

$h = 0$

$k = 2$

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.

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

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 $- 1 (- 1)$ .

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

Step 3.5.3.1.2

Move the negative in front of the fraction.

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

Step 3.5.3.2

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

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

Step 3.5.3.3

Divide $4$ by $4$ .

$- \frac{1}{1}$

Step 3.5.3.4

Cancel the common factor of $1$ .

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Step 3.5.3.4.1

Cancel the common factor.

$- \frac{1}{1}$

Step 3.5.3.4.2

Rewrite the expression.

$- 1 \cdot 1$

Step 3.5.3.5

Multiply $- 1$ by $1$ .

$- 1$

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.

$(- 1, 2)$

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.

$x = 1$

Step 3.9

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

Direction: Opens Left

Vertex: $(0, 2)$

Focus: $(- 1, 2)$

Axis of Symmetry: $y = 2$

Directrix: $x = 1$

Direction: Opens Left

Vertex: $(0, 2)$

Focus: $(- 1, 2)$

Axis of Symmetry: $y = 2$

Directrix: $x = 1$

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 $f (x) = 2 + 2 \sqrt{- 1 x}$ . In this case, the point is $(- 2, 4.82842712)$ .

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Step 4.1.1

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

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

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Multiply $- 1$ by $- 2$ .

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

Step 4.1.2.2

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

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

Step 4.1.3

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

$= 4.82842712$

Step 4.2

Substitute the $x$ value $- 2$ into $f (x) = 2 - 2 \sqrt{- 1 x}$ . In this case, the point is $(- 2, - 0.82842712)$ .

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Step 4.2.1

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

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

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Multiply $- 1$ by $- 2$ .

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

Step 4.2.2.2

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

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

Step 4.2.3

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

$= - 0.82842712$

Step 4.3

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

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Step 4.3.1

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

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

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 $- 1$ .

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

Step 4.3.2.1.2

Any root of $1$ is $1$ .

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

Step 4.3.2.1.3

Multiply $2$ by $1$ .

$f (- 1) = 2 + 2$

Step 4.3.2.2

Add $2$ and $2$ .

$f (- 1) = 4$

Step 4.3.2.3

The final answer is $4$ .

$y = 4$

Step 4.3.3

Convert $4$ to decimal.

$= 4$

Step 4.4

Substitute the $x$ value $- 1$ into $f (x) = 2 - 2 \sqrt{- 1 x}$ . In this case, the point is $(- 1, 0)$ .

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Step 4.4.1

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

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

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 $- 1$ .

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

Step 4.4.2.1.2

Any root of $1$ is $1$ .

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

Step 4.4.2.1.3

Multiply $- 2$ by $1$ .

$f (- 1) = 2 - 2$

Step 4.4.2.2

Subtract $2$ from $2$ .

$f (- 1) = 0$

Step 4.4.2.3

The final answer is $0$ .

$y = 0$

Step 4.4.3

Convert $0$ to decimal.

$= 0$

Step 4.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & 4.83 \\ - 2 & - 0.83 \\ - 1 & 4 \\ - 1 & 0 \\ 0 & 2 \end{array}$

Step 5

Graph the parabola using its properties and the selected points.

Direction: Opens Left

Vertex: $(0, 2)$

Focus: $(- 1, 2)$

Axis of Symmetry: $y = 2$

Directrix: $x = 1$

$\begin{array}{c} x & y \\ - 2 & 4.83 \\ - 2 & - 0.83 \\ - 1 & 4 \\ - 1 & 0 \\ 0 & 2 \end{array}$

Step 6