Algebra Examples

$x = {(y + 2)}^{2}$

Step 1

Simplify ${(y + 2)}^{2}$ .

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Step 1.1

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

$x = (y + 2) (y + 2)$

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

$x = y (y + 2) + 2 (y + 2)$

Step 1.2.2

Apply the distributive property.

$x = y \cdot y + y \cdot 2 + 2 (y + 2)$

Step 1.2.3

Apply the distributive property.

$x = y \cdot y + y \cdot 2 + 2 y + 2 \cdot 2$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $y$ by $y$ .

$x = y^{2} + y \cdot 2 + 2 y + 2 \cdot 2$

Step 1.3.1.2

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

$x = y^{2} + 2 \cdot y + 2 y + 2 \cdot 2$

Step 1.3.1.3

Multiply $2$ by $2$ .

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

Step 1.3.2

Add $2 y$ and $2 y$ .

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

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 $y^{2} + 4 y + 4$ .

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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 $c$ .

$a = 1$

$b = 4$

$c = 4$

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

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

Step 2.1.1.3.2

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

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Step 2.1.1.3.2.1

Factor $2$ out of $4$ .

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

Step 2.1.1.3.2.2

Cancel the common factors.

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Step 2.1.1.3.2.2.1

Factor $2$ out of $2 \cdot 1$ .

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

Step 2.1.1.3.2.2.2

Cancel the common factor.

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

Step 2.1.1.3.2.2.3

Rewrite the expression.

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

Step 2.1.1.3.2.2.4

Divide $2$ by $1$ .

$d = 2$

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

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

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

Cancel the common factor of $4^{2}$ and $4$ .

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Step 2.1.1.4.2.1.1.1

Factor $4$ out of $4^{2}$ .

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

Step 2.1.1.4.2.1.1.2

Cancel the common factors.

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Step 2.1.1.4.2.1.1.2.1

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

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

Step 2.1.1.4.2.1.1.2.2

Cancel the common factor.

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

Step 2.1.1.4.2.1.1.2.3

Rewrite the expression.

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

Step 2.1.1.4.2.1.1.2.4

Divide $4$ by $1$ .

$e = 4 - 1 \cdot 4$

Step 2.1.1.4.2.1.2

Multiply $- 1$ by $4$ .

$e = 4 - 4$

Step 2.1.1.4.2.2

Subtract $4$ from $4$ .

$e = 0$

Step 2.1.1.5

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

${(y + 2)}^{2} + 0$

Step 2.1.2

Set $x$ equal to the new right side.

$x = {(y + 2)}^{2} + 0$

Step 2.2

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

$a = 1$

$h = 0$

$k = - 2$

Step 2.3

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

Opens Right

Step 2.4

Find the vertex $(h, k)$ .

$(0, - 2)$

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.

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

Step 2.5.3

Cancel the common factor of $1$ .

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Step 2.5.3.1

Cancel the common factor.

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

Step 2.5.3.2

Rewrite the expression.

$\frac{1}{4}$

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.

$(\frac{1}{4}, - 2)$

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.

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

Step 2.9

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

Direction: Opens Right

Vertex: $(0, - 2)$

Focus: $(\frac{1}{4}, - 2)$

Axis of Symmetry: $y = - 2$

Directrix: $x = - \frac{1}{4}$

Direction: Opens Right

Vertex: $(0, - 2)$

Focus: $(\frac{1}{4}, - 2)$

Axis of Symmetry: $y = - 2$

Directrix: $x = - \frac{1}{4}$

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

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Step 3.1.1

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

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

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Remove parentheses.

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

Step 3.1.2.2

Any root of $1$ is $1$ .

$f (1) = 1 - 2$

Step 3.1.2.3

Subtract $2$ from $1$ .

$f (1) = - 1$

Step 3.1.2.4

The final answer is $- 1$ .

$y = - 1$

Step 3.1.3

Convert $- 1$ to decimal.

$= - 1$

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Simplify the result.

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Step 3.2.2.1

Remove parentheses.

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

Step 3.2.2.2

Simplify each term.

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Step 3.2.2.2.1

Any root of $1$ is $1$ .

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

Step 3.2.2.2.2

Multiply $- 1$ by $1$ .

$f (1) = - 1 - 2$

Step 3.2.2.3

Subtract $2$ from $- 1$ .

$f (1) = - 3$

Step 3.2.2.4

The final answer is $- 3$ .

$y = - 3$

Step 3.2.3

Convert $- 3$ to decimal.

$= - 3$

Step 3.3

Substitute the $x$ value $2$ into $f (x) = \sqrt{x} - 2$ . In this case, the point is $(2, - 0.58578643)$ .

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Step 3.3.1

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

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

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Remove parentheses.

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

Step 3.3.2.2

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

$y = \sqrt{2} - 2$

Step 3.3.3

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

$= - 0.58578643$

Step 3.4

Substitute the $x$ value $2$ into $f (x) = - \sqrt{x} - 2$ . In this case, the point is $(2, - 3.41421356)$ .

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Step 3.4.1

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

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

Step 3.4.2

Simplify the result.

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Step 3.4.2.1

Remove parentheses.

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

Step 3.4.2.2

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

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

Step 3.4.3

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

$= - 3.41421356$

Step 3.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & - 2 \\ 1 & - 1 \\ 1 & - 3 \\ 2 & - 0.59 \\ 2 & - 3.41 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(0, - 2)$

Focus: $(\frac{1}{4}, - 2)$

Axis of Symmetry: $y = - 2$

Directrix: $x = - \frac{1}{4}$

$\begin{array}{c} x & y \\ 0 & - 2 \\ 1 & - 1 \\ 1 & - 3 \\ 2 & - 0.59 \\ 2 & - 3.41 \end{array}$

Step 5