Algebra Examples

$y = {(x - 1)}^{2} - 5$

Step 1

Find the properties of the given parabola.

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Step 1.1

Use the vertex form,

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

$a$ ,

$h$ , and

$k$ .

$a = 1$

$h = 1$

$k = - 5$

Step 1.2

Since the value of

$a$ is positive, the parabola opens up.

Opens Up

Step 1.3

Find the vertex

$(h, k)$ .

$(1, - 5)$

Step 1.4

Find

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

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Step 1.4.1

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

$\frac{1}{4 a}$

Step 1.4.2

Substitute the value of

$a$ into the formula.

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

Step 1.4.3

Cancel the common factor of

$1$ .

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Step 1.4.3.1

Cancel the common factor.

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

Step 1.4.3.2

Rewrite the expression.

$\frac{1}{4}$

Step 1.5

Find the focus.

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Step 1.5.1

The focus of a parabola can be found by adding

$p$ to the y-coordinate

$k$ if the parabola opens up or down.

$(h, k + p)$

Step 1.5.2

Substitute the known values of

$h$ ,

$p$ , and

$k$ into the formula and simplify.

$(1, - \frac{19}{4})$

Step 1.6

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

$x = 1$

Step 1.7

Find the directrix.

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Step 1.7.1

The directrix of a parabola is the horizontal line found by subtracting

$p$ from the y-coordinate

$k$ of the vertex if the parabola opens up or down.

$y = k - p$

Step 1.7.2

Substitute the known values of

$p$ and

$k$ into the formula and simplify.

$y = - \frac{21}{4}$

Step 1.8

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

Direction: Opens Up

Vertex:

$(1, - 5)$

Focus:

$(1, - \frac{19}{4})$

Axis of Symmetry:

$x = 1$

Directrix:

$y = - \frac{21}{4}$

Direction: Opens Up

Vertex:

$(1, - 5)$

Focus:

$(1, - \frac{19}{4})$

Axis of Symmetry:

$x = 1$

Directrix:

$y = - \frac{21}{4}$

Step 2

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 2.1

Replace the variable

$x$ with

$0$ in the expression.

$f (0) = {(0)}^{2} - 2 \cdot 0 - 4$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Raising

$0$ to any positive power yields

$0$ .

$f (0) = 0 - 2 \cdot 0 - 4$

Step 2.2.1.2

Multiply

$- 2$ by

$0$ .

$f (0) = 0 + 0 - 4$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Add

$0$ and

$0$ .

$f (0) = 0 - 4$

Step 2.2.2.2

Subtract

$4$ from

$0$ .

$f (0) = - 4$

Step 2.2.3

The final answer is

$- 4$ .

$- 4$

Step 2.3

The

$y$ value at

$x = 0$ is

$- 4$ .

$y = - 4$

Step 2.4

Replace the variable

$x$ with

$- 1$ in the expression.

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

Step 2.5

Simplify the result.

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

Raise

$- 1$ to the power of

$2$ .

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

Step 2.5.1.2

Multiply

$- 2$ by

$- 1$ .

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

Step 2.5.2

Simplify by adding and subtracting.

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Step 2.5.2.1

Add

$1$ and

$2$ .

$f (- 1) = 3 - 4$

Step 2.5.2.2

Subtract

$4$ from

$3$ .

$f (- 1) = - 1$

Step 2.5.3

The final answer is

$- 1$ .

$- 1$

Step 2.6

The

$y$ value at

$x = - 1$ is

$- 1$ .

$y = - 1$

Step 2.7

Replace the variable

$x$ with

$2$ in the expression.

$f (2) = {(2)}^{2} - 2 \cdot 2 - 4$

Step 2.8

Simplify the result.

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Step 2.8.1

Simplify each term.

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Step 2.8.1.1

Raise

$2$ to the power of

$2$ .

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

Step 2.8.1.2

Multiply

$- 2$ by

$2$ .

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

Step 2.8.2

Simplify by subtracting numbers.

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Step 2.8.2.1

Subtract

$4$ from

$4$ .

$f (2) = 0 - 4$

Step 2.8.2.2

Subtract

$4$ from

$0$ .

$f (2) = - 4$

Step 2.8.3

The final answer is

$- 4$ .

$- 4$

Step 2.9

The

$y$ value at

$x = 2$ is

$- 4$ .

$y = - 4$

Step 2.10

Replace the variable

$x$ with

$3$ in the expression.

$f (3) = {(3)}^{2} - 2 \cdot 3 - 4$

Step 2.11

Simplify the result.

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Step 2.11.1

Simplify each term.

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Step 2.11.1.1

Raise

$3$ to the power of

$2$ .

$f (3) = 9 - 2 \cdot 3 - 4$

Step 2.11.1.2

Multiply

$- 2$ by

$3$ .

$f (3) = 9 - 6 - 4$

Step 2.11.2

Simplify by subtracting numbers.

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Step 2.11.2.1

Subtract

$6$ from

$9$ .

$f (3) = 3 - 4$

Step 2.11.2.2

Subtract

$4$ from

$3$ .

$f (3) = - 1$

Step 2.11.3

The final answer is

$- 1$ .

$- 1$

Step 2.12

The

$y$ value at

$x = 3$ is

$- 1$ .

$y = - 1$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & - 1 \\ 0 & - 4 \\ 1 & - 5 \\ 2 & - 4 \\ 3 & - 1 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex:

$(1, - 5)$

Focus:

$(1, - \frac{19}{4})$

Axis of Symmetry:

$x = 1$

Directrix:

$y = - \frac{21}{4}$

$\begin{array}{c} x & y \\ - 1 & - 1 \\ 0 & - 4 \\ 1 & - 5 \\ 2 & - 4 \\ 3 & - 1 \end{array}$

Step 4