Algebra Examples

y = (x + 2) (x - 3)

$y = (x + 2) (x - 3)$

Step 1

Find the properties of the given parabola.

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Step 1.1

Rewrite the equation in vertex form.

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Step 1.1.1

Complete the square for

(x + 2) (x - 3)

$(x + 2) (x - 3)$ .

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Step 1.1.1.1

Simplify the expression.

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Step 1.1.1.1.1

Expand

(x + 2) (x - 3)

$(x + 2) (x - 3)$ using the FOIL Method.

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Step 1.1.1.1.1.1

Apply the distributive property.

x (x - 3) + 2 (x - 3)

$x (x - 3) + 2 (x - 3)$

Step 1.1.1.1.1.2

Apply the distributive property.

x \cdot x + x \cdot - 3 + 2 (x - 3)

$x \cdot x + x \cdot - 3 + 2 (x - 3)$

Step 1.1.1.1.1.3

Apply the distributive property.

x \cdot x + x \cdot - 3 + 2 x + 2 \cdot - 3

$x \cdot x + x \cdot - 3 + 2 x + 2 \cdot - 3$

x \cdot x + x \cdot - 3 + 2 x + 2 \cdot - 3

$x \cdot x + x \cdot - 3 + 2 x + 2 \cdot - 3$

Step 1.1.1.1.2

Simplify and combine like terms.

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Step 1.1.1.1.2.1

Simplify each term.

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Step 1.1.1.1.2.1.1

Multiply

x

$x$ by

x

$x$ .

x^{2} + x \cdot - 3 + 2 x + 2 \cdot - 3

$x^{2} + x \cdot - 3 + 2 x + 2 \cdot - 3$

Step 1.1.1.1.2.1.2

Move

- 3

$- 3$ to the left of

x

$x$ .

x^{2} - 3 \cdot x + 2 x + 2 \cdot - 3

$x^{2} - 3 \cdot x + 2 x + 2 \cdot - 3$

Step 1.1.1.1.2.1.3

Multiply

2

$2$ by

- 3

$- 3$ .

x^{2} - 3 x + 2 x - 6

$x^{2} - 3 x + 2 x - 6$

x^{2} - 3 x + 2 x - 6

$x^{2} - 3 x + 2 x - 6$

Step 1.1.1.1.2.2

Add

- 3 x

$- 3 x$ and

2 x

$2 x$ .

x^{2} - x - 6

$x^{2} - x - 6$

x^{2} - x - 6

$x^{2} - x - 6$

x^{2} - x - 6

$x^{2} - x - 6$

Step 1.1.1.2

Use the form

a x^{2} + b x + c

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

a

$a$ ,

b

$b$ , and

c

$c$ .

a = 1

$a = 1$

b = - 1

$b = - 1$

c = - 6

$c = - 6$

Step 1.1.1.3

Consider the vertex form of a parabola.

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

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

Step 1.1.1.4

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

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

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Step 1.1.1.4.1

Substitute the values of

a

$a$ and

b

$b$ into the formula

d = \frac{b}{2 a}

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

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

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

Step 1.1.1.4.2

Simplify the right side.

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Step 1.1.1.4.2.1

Cancel the common factor of

- 1

$- 1$ and

1

$1$ .

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Step 1.1.1.4.2.1.1

Rewrite

- 1

$- 1$ as

- 1 (1)

$- 1 (1)$ .

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

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

Step 1.1.1.4.2.1.2

Cancel the common factor.

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

Step 1.1.1.4.2.1.3

Rewrite the expression.

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

Step 1.1.1.4.2.2

Move the negative in front of the fraction.

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

Step 1.1.1.5

Find the value of

$e$ using the formula

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

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Step 1.1.1.5.1

Substitute the values of

$c$ ,

$b$ and

$a$ into the formula

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

$e = - 6 - \frac{{(- 1)}^{2}}{4 \cdot 1}$

Step 1.1.1.5.2

Simplify the right side.

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Step 1.1.1.5.2.1

Simplify each term.

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Step 1.1.1.5.2.1.1

Raise

$- 1$ to the power of

$2$ .

$e = - 6 - \frac{1}{4 \cdot 1}$

Step 1.1.1.5.2.1.2

Multiply

$4$ by

$1$ .

$e = - 6 - \frac{1}{4}$

Step 1.1.1.5.2.2

To write

$- 6$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

$e = - 6 \cdot \frac{4}{4} - \frac{1}{4}$

Step 1.1.1.5.2.3

Combine

$- 6$ and

$\frac{4}{4}$ .

$e = \frac{- 6 \cdot 4}{4} - \frac{1}{4}$

Step 1.1.1.5.2.4

Combine the numerators over the common denominator.

$e = \frac{- 6 \cdot 4 - 1}{4}$

Step 1.1.1.5.2.5

Simplify the numerator.

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Step 1.1.1.5.2.5.1

Multiply

$- 6$ by

$4$ .

$e = \frac{- 24 - 1}{4}$

Step 1.1.1.5.2.5.2

Subtract

$1$ from

$- 24$ .

$e = \frac{- 25}{4}$

Step 1.1.1.5.2.6

Move the negative in front of the fraction.

$e = - \frac{25}{4}$

Step 1.1.1.6

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

${(x - \frac{1}{2})}^{2} - \frac{25}{4}$ .

${(x - \frac{1}{2})}^{2} - \frac{25}{4}$

Step 1.1.2

Set

$y$ equal to the new right side.

$y = {(x - \frac{1}{2})}^{2} - \frac{25}{4}$

Step 1.2

Use the vertex form,

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

$a$ ,

$h$ , and

$k$ .

$a = 1$

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

$k = - \frac{25}{4}$

Step 1.3

Since the value of

$a$ is positive, the parabola opens up.

Opens Up

Step 1.4

Find the vertex

$(h, k)$ .

$(\frac{1}{2}, - \frac{25}{4})$

Step 1.5

Find

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

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Step 1.5.1

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

$\frac{1}{4 a}$

Step 1.5.2

Substitute the value of

$a$ into the formula.

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

Step 1.5.3

Cancel the common factor of

$1$ .

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Step 1.5.3.1

Cancel the common factor.

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

Step 1.5.3.2

Rewrite the expression.

$\frac{1}{4}$

Step 1.6

Find the focus.

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Step 1.6.1

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 1.6.2

Substitute the known values of

$h$ ,

$p$ , and

$k$ into the formula and simplify.

$(\frac{1}{2}, - 6)$

Step 1.7

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

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

Step 1.8

Find the directrix.

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Step 1.8.1

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 1.8.2

Substitute the known values of

$p$ and

$k$ into the formula and simplify.

$y = - \frac{13}{2}$

Step 1.9

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

Direction: Opens Up

Vertex:

$(\frac{1}{2}, - \frac{25}{4})$

Focus:

$(\frac{1}{2}, - 6)$

Axis of Symmetry:

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

Directrix:

$y = - \frac{13}{2}$

Direction: Opens Up

Vertex:

$(\frac{1}{2}, - \frac{25}{4})$

Focus:

$(\frac{1}{2}, - 6)$

Axis of Symmetry:

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

Directrix:

$y = - \frac{13}{2}$

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

$- 1$ in the expression.

$f (- 1) = ((- 1) + 2) ((- 1) - 3)$

Step 2.2

Simplify the result.

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Step 2.2.1

Add

$- 1$ and

$2$ .

$f (- 1) = 1 ((- 1) - 3)$

Step 2.2.2

Multiply

$(- 1) - 3$ by

$1$ .

$f (- 1) = (- 1) - 3$

Step 2.2.3

Subtract

$3$ from

$- 1$ .

$f (- 1) = - 4$

Step 2.2.4

The final answer is

$- 4$ .

$- 4$

Step 2.3

The

$y$ value at

$x = - 1$ is

$- 4$ .

$y = - 4$

Step 2.4

Replace the variable

$x$ with

$- 2$ in the expression.

$f (- 2) = ((- 2) + 2) ((- 2) - 3)$

Step 2.5

Simplify the result.

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Step 2.5.1

Add

$- 2$ and

$2$ .

$f (- 2) = 0 ((- 2) - 3)$

Step 2.5.2

Subtract

$3$ from

$- 2$ .

$f (- 2) = 0 \cdot - 5$

Step 2.5.3

Multiply

$0$ by

$- 5$ .

$f (- 2) = 0$

Step 2.5.4

The final answer is

$0$ .

$0$

Step 2.6

The

$y$ value at

$x = - 2$ is

$0$ .

$y = 0$

Step 2.7

Replace the variable

$x$ with

$1$ in the expression.

$f (1) = ((1) + 2) ((1) - 3)$

Step 2.8

Simplify the result.

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Step 2.8.1

Add

$1$ and

$2$ .

$f (1) = 3 ((1) - 3)$

Step 2.8.2

Subtract

$3$ from

$1$ .

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

Step 2.8.3

Multiply

$3$ by

$- 2$ .

$f (1) = - 6$

Step 2.8.4

The final answer is

$- 6$ .

$- 6$

Step 2.9

The

$y$ value at

$x = 1$ is

$- 6$ .

$y = - 6$

Step 2.10

Replace the variable

$x$ with

$2$ in the expression.

$f (2) = ((2) + 2) ((2) - 3)$

Step 2.11

Simplify the result.

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Step 2.11.1

Add

$2$ and

$2$ .

$f (2) = 4 ((2) - 3)$

Step 2.11.2

Subtract

$3$ from

$2$ .

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

Step 2.11.3

Multiply

$4$ by

$- 1$ .

$f (2) = - 4$

Step 2.11.4

The final answer is

$- 4$ .

$- 4$

Step 2.12

The

$y$ value at

$x = 2$ is

$- 4$ .

$y = - 4$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & 0 \\ - 1 & - 4 \\ \frac{1}{2} & - \frac{25}{4} \\ 1 & - 6 \\ 2 & - 4 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex:

$(\frac{1}{2}, - \frac{25}{4})$

Focus:

$(\frac{1}{2}, - 6)$

Axis of Symmetry:

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

Directrix:

$y = - \frac{13}{2}$

$\begin{array}{c} x & y \\ - 2 & 0 \\ - 1 & - 4 \\ \frac{1}{2} & - \frac{25}{4} \\ 1 & - 6 \\ 2 & - 4 \end{array}$

Step 4