Precalculus Examples

${(y + 3)}^{2} = 4 (x - 1)$

Step 1

Rewrite the equation as $4 (x - 1) = {(y + 3)}^{2}$ .

$4 (x - 1) = {(y + 3)}^{2}$

Step 2

Divide each term in $4 (x - 1) = {(y + 3)}^{2}$ by $4$ and simplify.

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Step 2.1

Divide each term in $4 (x - 1) = {(y + 3)}^{2}$ by $4$ .

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

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.

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

Step 2.2.1.2

Divide $x - 1$ by $1$ .

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

Step 3

Add $1$ to both sides of the equation.

$x = \frac{{(y + 3)}^{2}}{4} + 1$

Step 4

Find the properties of the given parabola.

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Step 4.1

Reorder terms.

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

Step 4.2

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

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

$h = 1$

$k = - 3$

Step 4.3

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

Opens Right

Step 4.4

Find the vertex $(h, k)$ .

$(1, - 3)$

Step 4.5

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

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Step 4.5.1

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

$\frac{1}{4 a}$

Step 4.5.2

Substitute the value of $a$ into the formula.

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

Step 4.5.3

Simplify.

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Step 4.5.3.1

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

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

Step 4.5.3.2

Simplify by dividing numbers.

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Step 4.5.3.2.1

Divide $4$ by $4$ .

$\frac{1}{1}$

Step 4.5.3.2.2

Divide $1$ by $1$ .

$1$

Step 4.6

Find the focus.

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Step 4.6.1

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 4.6.2

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

$(2, - 3)$

Step 4.7

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

$y = - 3$

Step 4.8

Find the directrix.

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Step 4.8.1

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 4.8.2

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

$x = 0$

Step 4.9

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

Direction: Opens Right

Vertex: $(1, - 3)$

Focus: $(2, - 3)$

Axis of Symmetry: $y = - 3$

Directrix: $x = 0$

Direction: Opens Right

Vertex: $(1, - 3)$

Focus: $(2, - 3)$

Axis of Symmetry: $y = - 3$

Directrix: $x = 0$

Step 5

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 5.1

Substitute the $x$ value $2$ into $f (x) = 2 \sqrt{x - 1} - 3$ . In this case, the point is $(2, - 1)$ .

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Step 5.1.1

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

$f (2) = 2 \sqrt{(2) - 1} - 3$

Step 5.1.2

Simplify the result.

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Step 5.1.2.1

Simplify each term.

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Step 5.1.2.1.1

Subtract $1$ from $2$ .

$f (2) = 2 \sqrt{1} - 3$

Step 5.1.2.1.2

Any root of $1$ is $1$ .

$f (2) = 2 \cdot 1 - 3$

Step 5.1.2.1.3

Multiply $2$ by $1$ .

$f (2) = 2 - 3$

Step 5.1.2.2

Subtract $3$ from $2$ .

$f (2) = - 1$

Step 5.1.2.3

The final answer is $- 1$ .

$y = - 1$

Step 5.1.3

Convert $- 1$ to decimal.

$= - 1$

Step 5.2

Substitute the $x$ value $2$ into $f (x) = - 2 \sqrt{x - 1} - 3$ . In this case, the point is $(2, - 5)$ .

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Step 5.2.1

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

$f (2) = - 2 \sqrt{(2) - 1} - 3$

Step 5.2.2

Simplify the result.

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Step 5.2.2.1

Simplify each term.

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Step 5.2.2.1.1

Subtract $1$ from $2$ .

$f (2) = - 2 \sqrt{1} - 3$

Step 5.2.2.1.2

Any root of $1$ is $1$ .

$f (2) = - 2 \cdot 1 - 3$

Step 5.2.2.1.3

Multiply $- 2$ by $1$ .

$f (2) = - 2 - 3$

Step 5.2.2.2

Subtract $3$ from $- 2$ .

$f (2) = - 5$

Step 5.2.2.3

The final answer is $- 5$ .

$y = - 5$

Step 5.2.3

Convert $- 5$ to decimal.

$= - 5$

Step 5.3

Substitute the $x$ value $3$ into $f (x) = 2 \sqrt{x - 1} - 3$ . In this case, the point is $(3, - 0.17157287)$ .

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Step 5.3.1

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

$f (3) = 2 \sqrt{(3) - 1} - 3$

Step 5.3.2

Simplify the result.

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Step 5.3.2.1

Subtract $1$ from $3$ .

$f (3) = 2 \sqrt{2} - 3$

Step 5.3.2.2

The final answer is $2 \sqrt{2} - 3$ .

$y = 2 \sqrt{2} - 3$

Step 5.3.3

Convert $2 \sqrt{2} - 3$ to decimal.

$= - 0.17157287$

Step 5.4

Substitute the $x$ value $3$ into $f (x) = - 2 \sqrt{x - 1} - 3$ . In this case, the point is $(3, - 5.82842712)$ .

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Step 5.4.1

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

$f (3) = - 2 \sqrt{(3) - 1} - 3$

Step 5.4.2

Simplify the result.

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Step 5.4.2.1

Subtract $1$ from $3$ .

$f (3) = - 2 \sqrt{2} - 3$

Step 5.4.2.2

The final answer is $- 2 \sqrt{2} - 3$ .

$y = - 2 \sqrt{2} - 3$

Step 5.4.3

Convert $- 2 \sqrt{2} - 3$ to decimal.

$= - 5.82842712$

Step 5.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 1 & - 3 \\ 2 & - 1 \\ 2 & - 5 \\ 3 & - 0.17 \\ 3 & - 5.83 \end{array}$

Step 6

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(1, - 3)$

Focus: $(2, - 3)$

Axis of Symmetry: $y = - 3$

Directrix: $x = 0$

$\begin{array}{c} x & y \\ 1 & - 3 \\ 2 & - 1 \\ 2 & - 5 \\ 3 & - 0.17 \\ 3 & - 5.83 \end{array}$

Step 7