Rewrite the equation in vertex form.

Complete the square for <math><mstyle displaystyle="true"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>11</mn><mi>x</mi><mo>-</mo><mn>26</mn></mstyle></math> .

Use the form <math><mstyle displaystyle="true"><mi>a</mi><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mstyle></math> , to find the values of <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> , <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> , and <math><mstyle displaystyle="true"><mi>c</mi></mstyle></math> .

Consider the vertex form of a parabola.

Substitute the values of <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> and <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> into the formula <math><mstyle displaystyle="true"><mi>d</mi><mo>=</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Find the value of <math><mstyle displaystyle="true"><mi>e</mi></mstyle></math> using the formula <math><mstyle displaystyle="true"><mi>e</mi><mo>=</mo><mi>c</mi><mo>-</mo><mfrac><mrow><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>4</mn><mi>a</mi></mrow></mfrac></mstyle></math> .

Simplify each term.

Raise <math><mstyle displaystyle="true"><mo>-</mo><mn>11</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

To write <math><mstyle displaystyle="true"><mo>-</mo><mn>26</mn></mstyle></math> as a fraction with a common denominator, multiply by <math><mstyle displaystyle="true"><mfrac><mrow><mn>4</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> .

Combine <math><mstyle displaystyle="true"><mo>-</mo><mn>26</mn></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mn>4</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>26</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>121</mn></mstyle></math> from <math><mstyle displaystyle="true"><mo>-</mo><mn>104</mn></mstyle></math> .

Move the negative in front of the fraction.

Substitute the values of <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> , <math><mstyle displaystyle="true"><mi>d</mi></mstyle></math> , and <math><mstyle displaystyle="true"><mi>e</mi></mstyle></math> into the vertex form <math><mstyle displaystyle="true"><mi>a</mi><msup><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mi>d</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>e</mi></mstyle></math> .

Set <math><mstyle displaystyle="true"><mi>y</mi></mstyle></math> equal to the new right side.

Use the vertex form, <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mi>a</mi><msup><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mi>h</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>k</mi></mstyle></math> , to determine the values of <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> , <math><mstyle displaystyle="true"><mi>h</mi></mstyle></math> , and <math><mstyle displaystyle="true"><mi>k</mi></mstyle></math> .

Since the value of <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> is positive, the parabola opens up.

Opens Up

Find the vertex <math><mstyle displaystyle="true"><mrow><mo>(</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></mstyle></math> .

Find <math><mstyle displaystyle="true"><mi>p</mi></mstyle></math> , the distance from the vertex to the focus.

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

Substitute the value of <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> into the formula.

Cancel the common factor of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Find the focus.

The focus of a parabola can be found by adding <math><mstyle displaystyle="true"><mi>p</mi></mstyle></math> to the y-coordinate <math><mstyle displaystyle="true"><mi>k</mi></mstyle></math> if the parabola opens up or down.

Substitute the known values of <math><mstyle displaystyle="true"><mi>h</mi></mstyle></math> , <math><mstyle displaystyle="true"><mi>p</mi></mstyle></math> , and <math><mstyle displaystyle="true"><mi>k</mi></mstyle></math> into the formula and simplify.

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

Find the directrix.

The directrix of a parabola is the horizontal line found by subtracting <math><mstyle displaystyle="true"><mi>p</mi></mstyle></math> from the y-coordinate <math><mstyle displaystyle="true"><mi>k</mi></mstyle></math> of the vertex if the parabola opens up or down.

Substitute the known values of <math><mstyle displaystyle="true"><mi>p</mi></mstyle></math> and <math><mstyle displaystyle="true"><mi>k</mi></mstyle></math> into the formula and simplify.

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

Direction: Opens Up

Vertex: <math><mstyle displaystyle="true"><mrow><mo>(</mo><mfrac><mrow><mn>11</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mo>-</mo><mfrac><mrow><mn>225</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math>

Focus: <math><mstyle displaystyle="true"><mrow><mo>(</mo><mfrac><mrow><mn>11</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mo>-</mo><mn>56</mn><mo>)</mo></mrow></mstyle></math>

Axis of Symmetry: <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mfrac><mrow><mn>11</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>

Directrix: <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>113</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>

Direction: Opens Up

Vertex: <math><mstyle displaystyle="true"><mrow><mo>(</mo><mfrac><mrow><mn>11</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mo>-</mo><mfrac><mrow><mn>225</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math>

Focus: <math><mstyle displaystyle="true"><mrow><mo>(</mo><mfrac><mrow><mn>11</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mo>-</mo><mn>56</mn><mo>)</mo></mrow></mstyle></math>

Axis of Symmetry: <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mfrac><mrow><mn>11</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>

Directrix: <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>113</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>

Replace the variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mn>5</mn></mstyle></math> in the expression.

Simplify the result.

Simplify each term.

Raise <math><mstyle displaystyle="true"><mn>5</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>11</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>5</mn></mstyle></math> .

Simplify by subtracting numbers.

Subtract <math><mstyle displaystyle="true"><mn>55</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>25</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>26</mn></mstyle></math> from <math><mstyle displaystyle="true"><mo>-</mo><mn>30</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><mo>-</mo><mn>56</mn></mstyle></math> .

The <math><mstyle displaystyle="true"><mi>y</mi></mstyle></math> value at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mn>5</mn></mstyle></math> is <math><mstyle displaystyle="true"><mo>-</mo><mn>56</mn></mstyle></math> .

Replace the variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> in the expression.

Simplify the result.

Simplify each term.

Raise <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>11</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Simplify by subtracting numbers.

Subtract <math><mstyle displaystyle="true"><mn>44</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>16</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>26</mn></mstyle></math> from <math><mstyle displaystyle="true"><mo>-</mo><mn>28</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><mo>-</mo><mn>54</mn></mstyle></math> .

The <math><mstyle displaystyle="true"><mi>y</mi></mstyle></math> value at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mn>4</mn></mstyle></math> is <math><mstyle displaystyle="true"><mo>-</mo><mn>54</mn></mstyle></math> .

Replace the variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mn>7</mn></mstyle></math> in the expression.

Simplify the result.

Simplify each term.

Raise <math><mstyle displaystyle="true"><mn>7</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>11</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>7</mn></mstyle></math> .

Simplify by subtracting numbers.

Subtract <math><mstyle displaystyle="true"><mn>77</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>49</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>26</mn></mstyle></math> from <math><mstyle displaystyle="true"><mo>-</mo><mn>28</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><mo>-</mo><mn>54</mn></mstyle></math> .

The <math><mstyle displaystyle="true"><mi>y</mi></mstyle></math> value at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mn>7</mn></mstyle></math> is <math><mstyle displaystyle="true"><mo>-</mo><mn>54</mn></mstyle></math> .

Replace the variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mn>8</mn></mstyle></math> in the expression.

Simplify the result.

Simplify each term.

Raise <math><mstyle displaystyle="true"><mn>8</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>11</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>8</mn></mstyle></math> .

Simplify by subtracting numbers.

Subtract <math><mstyle displaystyle="true"><mn>88</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>64</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>26</mn></mstyle></math> from <math><mstyle displaystyle="true"><mo>-</mo><mn>24</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><mo>-</mo><mn>50</mn></mstyle></math> .

The <math><mstyle displaystyle="true"><mi>y</mi></mstyle></math> value at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mn>8</mn></mstyle></math> is <math><mstyle displaystyle="true"><mo>-</mo><mn>50</mn></mstyle></math> .

Graph the parabola using its properties and the selected points.

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: <math><mstyle displaystyle="true"><mrow><mo>(</mo><mfrac><mrow><mn>11</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mo>-</mo><mfrac><mrow><mn>225</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math>

Focus: <math><mstyle displaystyle="true"><mrow><mo>(</mo><mfrac><mrow><mn>11</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mo>-</mo><mn>56</mn><mo>)</mo></mrow></mstyle></math>

Axis of Symmetry: <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mfrac><mrow><mn>11</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>

Directrix: <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>113</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>

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Name | one billion six hundred ninety-one million one hundred twenty thousand eight hundred fifty-seven |
---|

- 1691120857 has 4 divisors, whose sum is
**1692155556** - The reverse of 1691120857 is
**7580211961** - Previous prime number is
**1637**

- Is Prime? no
- Number parity odd
- Number length 10
- Sum of Digits 40
- Digital Root 4

Name | two billion one hundred eight million two hundred twenty-eight thousand six hundred eighty-nine |
---|

- 2108228689 has 16 divisors, whose sum is
**2149353792** - The reverse of 2108228689 is
**9868228012** - Previous prime number is
**173**

- Is Prime? no
- Number parity odd
- Number length 10
- Sum of Digits 46
- Digital Root 1

Name | two hundred eighty-five million sixty thousand three hundred one |
---|

- 285060301 has 4 divisors, whose sum is
**301828572** - The reverse of 285060301 is
**103060582** - Previous prime number is
**17**

- Is Prime? no
- Number parity odd
- Number length 9
- Sum of Digits 25
- Digital Root 7