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

Subtract

$2 y$ from

$- 2 y$ .

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

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

Rewrite

$- 4$ as

$- 1 (4)$ .

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

Step 2.1.1.4.2.1.1.2

Apply the product rule to

$- 1 (4)$ .

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

Step 2.1.1.4.2.1.1.3

Raise

$- 1$ to the power of

$2$ .

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

Step 2.1.1.4.2.1.1.4

Multiply

$4^{2}$ by

$1$ .

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

Step 2.1.1.4.2.1.1.5

Factor

$4$ out of

$4^{2}$ .

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

Step 2.1.1.4.2.1.1.6

Cancel the common factors.

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Step 2.1.1.4.2.1.1.6.1

Factor

$4$ out of

$4 \cdot 1$ .

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

Step 2.1.1.4.2.1.1.6.2

Cancel the common factor.

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

Step 2.1.1.4.2.1.1.6.3

Rewrite the expression.

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

Step 2.1.1.4.2.1.1.6.4

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

$(1, 3)$ .

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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

Add

$1$ and

$2$ .

$f (1) = 3$

Step 3.1.2.4

The final answer is

$3$ .

$y = 3$

Step 3.1.3

Convert

$3$ to decimal.

$= 3$

Step 3.2

Substitute the

$x$ value

$1$ into

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

$(1, 1)$ .

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Step 3.2.1

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

Add

$- 1$ and

$2$ .

$f (1) = 1$

Step 3.2.2.4

The final answer is

$1$ .

$y = 1$

Step 3.2.3

Convert

$1$ to decimal.

$= 1$

Step 3.3

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 3.3.1

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.

$= 3.41421356$

Step 3.4

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 3.4.1

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.

$= 0.58578643$

Step 3.5

Graph the parabola using its properties and the selected points.

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

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:

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

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

Step 5