Algebra Examples

$f (x) = {(x - 7)}^{2}$

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 $k$ .

$a = 1$

$h = 7$

$k = 0$

Step 1.2

Since the value of $a$ is positive, the parabola opens up.

Opens Up

Step 1.3

Find the vertex $(h, k)$ .

$(7, 0)$

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.

$(7, \frac{1}{4})$

Step 1.6

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

$x = 7$

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.

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

Step 1.8

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

Direction: Opens Up

Vertex: $(7, 0)$

Focus: $(7, \frac{1}{4})$

Axis of Symmetry: $x = 7$

Directrix: $y = - \frac{1}{4}$

Direction: Opens Up

Vertex: $(7, 0)$

Focus: $(7, \frac{1}{4})$

Axis of Symmetry: $x = 7$

Directrix: $y = - \frac{1}{4}$

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 $6$ in the expression.

$f (6) = {(6)}^{2} - 14 \cdot 6 + 49$

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

Raise $6$ to the power of $2$ .

$f (6) = 36 - 14 \cdot 6 + 49$

Step 2.2.1.2

Multiply $- 14$ by $6$ .

$f (6) = 36 - 84 + 49$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Subtract $84$ from $36$ .

$f (6) = - 48 + 49$

Step 2.2.2.2

Add $- 48$ and $49$ .

$f (6) = 1$

Step 2.2.3

The final answer is $1$ .

$1$

Step 2.3

The $y$ value at $x = 6$ is $1$ .

$y = 1$

Step 2.4

Replace the variable $x$ with $5$ in the expression.

$f (5) = {(5)}^{2} - 14 \cdot 5 + 49$

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 $5$ to the power of $2$ .

$f (5) = 25 - 14 \cdot 5 + 49$

Step 2.5.1.2

Multiply $- 14$ by $5$ .

$f (5) = 25 - 70 + 49$

Step 2.5.2

Simplify by adding and subtracting.

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Step 2.5.2.1

Subtract $70$ from $25$ .

$f (5) = - 45 + 49$

Step 2.5.2.2

Add $- 45$ and $49$ .

$f (5) = 4$

Step 2.5.3

The final answer is $4$ .

$4$

Step 2.6

The $y$ value at $x = 5$ is $4$ .

$y = 4$

Step 2.7

Replace the variable $x$ with $8$ in the expression.

$f (8) = {(8)}^{2} - 14 \cdot 8 + 49$

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 $8$ to the power of $2$ .

$f (8) = 64 - 14 \cdot 8 + 49$

Step 2.8.1.2

Multiply $- 14$ by $8$ .

$f (8) = 64 - 112 + 49$

Step 2.8.2

Simplify by adding and subtracting.

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Step 2.8.2.1

Subtract $112$ from $64$ .

$f (8) = - 48 + 49$

Step 2.8.2.2

Add $- 48$ and $49$ .

$f (8) = 1$

Step 2.8.3

The final answer is $1$ .

$1$

Step 2.9

The $y$ value at $x = 8$ is $1$ .

$y = 1$

Step 2.10

Replace the variable $x$ with $9$ in the expression.

$f (9) = {(9)}^{2} - 14 \cdot 9 + 49$

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 $9$ to the power of $2$ .

$f (9) = 81 - 14 \cdot 9 + 49$

Step 2.11.1.2

Multiply $- 14$ by $9$ .

$f (9) = 81 - 126 + 49$

Step 2.11.2

Simplify by adding and subtracting.

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Step 2.11.2.1

Subtract $126$ from $81$ .

$f (9) = - 45 + 49$

Step 2.11.2.2

Add $- 45$ and $49$ .

$f (9) = 4$

Step 2.11.3

The final answer is $4$ .

$4$

Step 2.12

The $y$ value at $x = 9$ is $4$ .

$y = 4$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 5 & 4 \\ 6 & 1 \\ 7 & 0 \\ 8 & 1 \\ 9 & 4 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(7, 0)$

Focus: $(7, \frac{1}{4})$

Axis of Symmetry: $x = 7$

Directrix: $y = - \frac{1}{4}$

$\begin{array}{c} x & y \\ 5 & 4 \\ 6 & 1 \\ 7 & 0 \\ 8 & 1 \\ 9 & 4 \end{array}$

Step 4