Algebra Examples

y = (x - 1) (x - 4)

$y = (x - 1) (x - 4)$

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 - 1) (x - 4)

$(x - 1) (x - 4)$ .

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Step 1.1.1.1

Simplify the expression.

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Step 1.1.1.1.1

Expand

(x - 1) (x - 4)

$(x - 1) (x - 4)$ using the FOIL Method.

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Step 1.1.1.1.1.1

Apply the distributive property.

x (x - 4) - 1 (x - 4)

$x (x - 4) - 1 (x - 4)$

Step 1.1.1.1.1.2

Apply the distributive property.

x \cdot x + x \cdot - 4 - 1 (x - 4)

$x \cdot x + x \cdot - 4 - 1 (x - 4)$

Step 1.1.1.1.1.3

Apply the distributive property.

x \cdot x + x \cdot - 4 - 1 x - 1 \cdot - 4

$x \cdot x + x \cdot - 4 - 1 x - 1 \cdot - 4$

x \cdot x + x \cdot - 4 - 1 x - 1 \cdot - 4

$x \cdot x + x \cdot - 4 - 1 x - 1 \cdot - 4$

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 - 4 - 1 x - 1 \cdot - 4

$x^{2} + x \cdot - 4 - 1 x - 1 \cdot - 4$

Step 1.1.1.1.2.1.2

Move

- 4

$- 4$ to the left of

x

$x$ .

x^{2} - 4 \cdot x - 1 x - 1 \cdot - 4

$x^{2} - 4 \cdot x - 1 x - 1 \cdot - 4$

Step 1.1.1.1.2.1.3

Rewrite

- 1 x

$- 1 x$ as

- x

$- x$ .

x^{2} - 4 x - x - 1 \cdot - 4

$x^{2} - 4 x - x - 1 \cdot - 4$

Step 1.1.1.1.2.1.4

Multiply

- 1

$- 1$ by

- 4

$- 4$ .

x^{2} - 4 x - x + 4

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

x^{2} - 4 x - x + 4

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

Step 1.1.1.1.2.2

Subtract

x

$x$ from

- 4 x

$- 4 x$ .

x^{2} - 5 x + 4

$x^{2} - 5 x + 4$

x^{2} - 5 x + 4

$x^{2} - 5 x + 4$

x^{2} - 5 x + 4

$x^{2} - 5 x + 4$

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 = - 5

$b = - 5$

c = 4

$c = 4$

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{- 5}{2 \cdot 1}

$d = \frac{- 5}{2 \cdot 1}$

Step 1.1.1.4.2

Simplify the right side.

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Step 1.1.1.4.2.1

Multiply

2

$2$ by

1

$1$ .

d = \frac{- 5}{2}

$d = \frac{- 5}{2}$

Step 1.1.1.4.2.2

Move the negative in front of the fraction.

d = - \frac{5}{2}

$d = - \frac{5}{2}$

d = - \frac{5}{2}

$d = - \frac{5}{2}$

d = - \frac{5}{2}

$d = - \frac{5}{2}$

Step 1.1.1.5

Find the value of

e

$e$ using the formula

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

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

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Step 1.1.1.5.1

Substitute the values of

c

$c$ ,

b

$b$ and

a

$a$ into the formula

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

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

e = 4 - \frac{{(- 5)}^{2}}{4 \cdot 1}

$e = 4 - \frac{{(- 5)}^{2}}{4 \cdot 1}$

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

- 5

$- 5$ to the power of

2

$2$ .

e = 4 - \frac{25}{4 \cdot 1}

$e = 4 - \frac{25}{4 \cdot 1}$

Step 1.1.1.5.2.1.2

Multiply

4

$4$ by

1

$1$ .

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

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

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

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

Step 1.1.1.5.2.2

To write

4

$4$ as a fraction with a common denominator, multiply by

\frac{4}{4}

$\frac{4}{4}$ .

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

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

Step 1.1.1.5.2.3

Combine

4

$4$ and

\frac{4}{4}

$\frac{4}{4}$ .

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

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

Step 1.1.1.5.2.4

Combine the numerators over the common denominator.

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

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

Step 1.1.1.5.2.5

Simplify the numerator.

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Step 1.1.1.5.2.5.1

Multiply

4

$4$ by

4

$4$ .

e = \frac{16 - 25}{4}

$e = \frac{16 - 25}{4}$

Step 1.1.1.5.2.5.2

Subtract

25

$25$ from

16

$16$ .

e = \frac{- 9}{4}

$e = \frac{- 9}{4}$

e = \frac{- 9}{4}

$e = \frac{- 9}{4}$

Step 1.1.1.5.2.6

Move the negative in front of the fraction.

e = - \frac{9}{4}

$e = - \frac{9}{4}$

e = - \frac{9}{4}

$e = - \frac{9}{4}$

e = - \frac{9}{4}

$e = - \frac{9}{4}$

Step 1.1.1.6

Substitute the values of

a

$a$ ,

d

$d$ , and

e

$e$ into the vertex form

{(x - \frac{5}{2})}^{2} - \frac{9}{4}

${(x - \frac{5}{2})}^{2} - \frac{9}{4}$ .

{(x - \frac{5}{2})}^{2} - \frac{9}{4}

${(x - \frac{5}{2})}^{2} - \frac{9}{4}$

{(x - \frac{5}{2})}^{2} - \frac{9}{4}

${(x - \frac{5}{2})}^{2} - \frac{9}{4}$

Step 1.1.2

Set

y

$y$ equal to the new right side.

y = {(x - \frac{5}{2})}^{2} - \frac{9}{4}

$y = {(x - \frac{5}{2})}^{2} - \frac{9}{4}$

y = {(x - \frac{5}{2})}^{2} - \frac{9}{4}

$y = {(x - \frac{5}{2})}^{2} - \frac{9}{4}$

Step 1.2

Use the vertex form,

y = a {(x - h)}^{2} + k

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

a

$a$ ,

h

$h$ , and

k

$k$ .

a = 1

$a = 1$

h = \frac{5}{2}

$h = \frac{5}{2}$

k = - \frac{9}{4}

$k = - \frac{9}{4}$

Step 1.3

Since the value of

a

$a$ is positive, the parabola opens up.

Opens Up

Step 1.4

Find the vertex

(h, k)

$(h, k)$ .

(\frac{5}{2}, - \frac{9}{4})

$(\frac{5}{2}, - \frac{9}{4})$

Step 1.5

Find

p

$p$ , 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}

$\frac{1}{4 a}$

Step 1.5.2

Substitute the value of

a

$a$ into the formula.

\frac{1}{4 \cdot 1}

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

Step 1.5.3

Cancel the common factor of

1

$1$ .

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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.

$(\frac{5}{2}, - 2)$

Step 1.7

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

$x = \frac{5}{2}$

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.

$y = - \frac{5}{2}$

Step 1.9

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

Direction: Opens Up

Vertex:

$(\frac{5}{2}, - \frac{9}{4})$

Focus:

$(\frac{5}{2}, - 2)$

Axis of Symmetry:

$x = \frac{5}{2}$

Directrix:

$y = - \frac{5}{2}$

Direction: Opens Up

Vertex:

$(\frac{5}{2}, - \frac{9}{4})$

Focus:

$(\frac{5}{2}, - 2)$

Axis of Symmetry:

$x = \frac{5}{2}$

Directrix:

$y = - \frac{5}{2}$

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.

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

Step 2.2

Simplify the result.

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Step 2.2.1

Subtract

$1$ from

$1$ .

$f (1) = 0 ((1) - 4)$

Step 2.2.2

Subtract

$4$ from

$1$ .

$f (1) = 0 \cdot - 3$

Step 2.2.3

Multiply

$0$ by

$- 3$ .

$f (1) = 0$

Step 2.2.4

The final answer is

$0$ .

$0$

Step 2.3

The

$y$ value at

$x = 1$ is

$0$ .

$y = 0$

Step 2.4

Replace the variable

$x$ with

$0$ in the expression.

$f (0) = ((0) - 1) ((0) - 4)$

Step 2.5

Simplify the result.

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Step 2.5.1

Subtract

$1$ from

$0$ .

$f (0) = - 1 ((0) - 4)$

Step 2.5.2

Subtract

$4$ from

$0$ .

$f (0) = - 1 \cdot - 4$

Step 2.5.3

Multiply

$- 1$ by

$- 4$ .

$f (0) = 4$

Step 2.5.4

The final answer is

$4$ .

$4$

Step 2.6

The

$y$ value at

$x = 0$ is

$4$ .

$y = 4$

Step 2.7

Replace the variable

$x$ with

$3$ in the expression.

$f (3) = ((3) - 1) ((3) - 4)$

Step 2.8

Simplify the result.

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Step 2.8.1

Subtract

$1$ from

$3$ .

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

Step 2.8.2

Subtract

$4$ from

$3$ .

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

Step 2.8.3

Multiply

$2$ by

$- 1$ .

$f (3) = - 2$

Step 2.8.4

The final answer is

$- 2$ .

$- 2$

Step 2.9

The

$y$ value at

$x = 3$ is

$- 2$ .

$y = - 2$

Step 2.10

Replace the variable

$x$ with

$4$ in the expression.

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

Step 2.11

Simplify the result.

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Step 2.11.1

Subtract

$1$ from

$4$ .

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

Step 2.11.2

Subtract

$4$ from

$4$ .

$f (4) = 3 \cdot 0$

Step 2.11.3

Multiply

$3$ by

$0$ .

$f (4) = 0$

Step 2.11.4

The final answer is

$0$ .

$0$

Step 2.12

The

$y$ value at

$x = 4$ is

$0$ .

$y = 0$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & 4 \\ 1 & 0 \\ \frac{5}{2} & - \frac{9}{4} \\ 3 & - 2 \\ 4 & 0 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex:

$(\frac{5}{2}, - \frac{9}{4})$

Focus:

$(\frac{5}{2}, - 2)$

Axis of Symmetry:

$x = \frac{5}{2}$

Directrix:

$y = - \frac{5}{2}$

$\begin{array}{c} x & y \\ 0 & 4 \\ 1 & 0 \\ \frac{5}{2} & - \frac{9}{4} \\ 3 & - 2 \\ 4 & 0 \end{array}$

Step 4