Algebra Examples

$f (x) = x^{2} - 5 x + 4$

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

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

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Step 1.1.1.1

Use the form

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

$a$ ,

$b$ , and

$c$ .

$a = 1$

$b = - 5$

$c = 4$

Step 1.1.1.2

Consider the vertex form of a parabola.

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

Step 1.1.1.3

Find the value of

$d$ using the formula

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

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Step 1.1.1.3.1

Substitute the values of

$a$ and

$b$ into the formula

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

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

Step 1.1.1.3.2

Simplify the right side.

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Step 1.1.1.3.2.1

Multiply

$2$ by

$1$ .

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

Step 1.1.1.3.2.2

Move the negative in front of the fraction.

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

Step 1.1.1.4

Find the value of

$e$ using the formula

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

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Step 1.1.1.4.1

Substitute the values of

$c$ ,

$b$ and

$a$ into the formula

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

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

Step 1.1.1.4.2

Simplify the right side.

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Step 1.1.1.4.2.1

Simplify each term.

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Step 1.1.1.4.2.1.1

Raise

$- 5$ to the power of

$2$ .

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

Step 1.1.1.4.2.1.2

Multiply

$4$ by

$1$ .

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

Step 1.1.1.4.2.2

To write

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

$\frac{4}{4}$ .

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

Step 1.1.1.4.2.3

Combine

$4$ and

$\frac{4}{4}$ .

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

Step 1.1.1.4.2.4

Combine the numerators over the common denominator.

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

Step 1.1.1.4.2.5

Simplify the numerator.

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Step 1.1.1.4.2.5.1

Multiply

$4$ by

$4$ .

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

Step 1.1.1.4.2.5.2

Subtract

$25$ from

$16$ .

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

Step 1.1.1.4.2.6

Move the negative in front of the fraction.

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

Step 1.1.1.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

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

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

Step 1.1.2

Set

$y$ equal to the new right side.

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

Step 1.2

Use the vertex form,

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

$a$ ,

$h$ , and

$k$ .

$a = 1$

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

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

Step 1.3

Since the value of

$a$ is positive, the parabola opens up.

Opens Up

Step 1.4

Find the vertex

$(h, k)$ .

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

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.

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

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.

$f (1) = {(1)}^{2} - 5 \cdot 1 + 4$

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

One to any power is one.

$f (1) = 1 - 5 \cdot 1 + 4$

Step 2.2.1.2

Multiply

$- 5$ by

$1$ .

$f (1) = 1 - 5 + 4$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Subtract

$5$ from

$1$ .

$f (1) = - 4 + 4$

Step 2.2.2.2

Add

$- 4$ and

$4$ .

$f (1) = 0$

Step 2.2.3

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.

$f (0) = {(0)}^{2} - 5 \cdot 0 + 4$

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

Raising

$0$ to any positive power yields

$0$ .

$f (0) = 0 - 5 \cdot 0 + 4$

Step 2.5.1.2

Multiply

$- 5$ by

$0$ .

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

Step 2.5.2

Simplify by adding numbers.

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Step 2.5.2.1

Add

$0$ and

$0$ .

$f (0) = 0 + 4$

Step 2.5.2.2

Add

$0$ and

$4$ .

$f (0) = 4$

Step 2.5.3

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.

$f (3) = {(3)}^{2} - 5 \cdot 3 + 4$

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

$3$ to the power of

$2$ .

$f (3) = 9 - 5 \cdot 3 + 4$

Step 2.8.1.2

Multiply

$- 5$ by

$3$ .

$f (3) = 9 - 15 + 4$

Step 2.8.2

Simplify by adding and subtracting.

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Step 2.8.2.1

Subtract

$15$ from

$9$ .

$f (3) = - 6 + 4$

Step 2.8.2.2

Add

$- 6$ and

$4$ .

$f (3) = - 2$

Step 2.8.3

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.

$f (4) = {(4)}^{2} - 5 \cdot 4 + 4$

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

$4$ to the power of

$2$ .

$f (4) = 16 - 5 \cdot 4 + 4$

Step 2.11.1.2

Multiply

$- 5$ by

$4$ .

$f (4) = 16 - 20 + 4$

Step 2.11.2

Simplify by adding and subtracting.

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Step 2.11.2.1

Subtract

$20$ from

$16$ .

$f (4) = - 4 + 4$

Step 2.11.2.2

Add

$- 4$ and

$4$ .

$f (4) = 0$

Step 2.11.3

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