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><mo>-</mo><mn>3</mn><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mstyle></math>

Focus: <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mn>3</mn><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math>

Axis of Symmetry: <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mn>3</mn></mstyle></math>

Directrix: <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>

Direction: Opens Up

Vertex: <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mn>3</mn><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mstyle></math>

Focus: <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mn>3</mn><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math>

Axis of Symmetry: <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mn>3</mn></mstyle></math>

Directrix: <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>

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

Simplify the result.

Simplify each term.

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

Multiply <math><mstyle displaystyle="true"><mn>6</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>4</mn></mstyle></math> .

Simplify by adding and subtracting.

Subtract <math><mstyle displaystyle="true"><mn>24</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>16</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>8</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>9</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

The <math><mstyle displaystyle="true"><mi>y</mi></mstyle></math> value at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mn>4</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

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

Simplify the result.

Simplify each term.

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

Multiply <math><mstyle displaystyle="true"><mn>6</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>5</mn></mstyle></math> .

Simplify by adding and subtracting.

Subtract <math><mstyle displaystyle="true"><mn>30</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>25</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>5</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>9</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

The <math><mstyle displaystyle="true"><mi>y</mi></mstyle></math> value at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mn>5</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Replace the variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn></mstyle></math> in the expression.

Simplify the result.

Simplify each term.

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

Multiply <math><mstyle displaystyle="true"><mn>6</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn></mstyle></math> .

Simplify by adding and subtracting.

Subtract <math><mstyle displaystyle="true"><mn>12</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>8</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>9</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

The <math><mstyle displaystyle="true"><mi>y</mi></mstyle></math> value at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mn>2</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Replace the variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> in the expression.

Simplify the result.

Simplify each term.

Raise <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>6</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

Simplify by adding and subtracting.

Subtract <math><mstyle displaystyle="true"><mn>6</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>5</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>9</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

The <math><mstyle displaystyle="true"><mi>y</mi></mstyle></math> value at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>4</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><mo>-</mo><mn>3</mn><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mstyle></math>

Focus: <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mn>3</mn><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math>

Axis of Symmetry: <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mn>3</mn></mstyle></math>

Directrix: <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>

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Name | two hundred twenty-five million eight hundred twenty-nine thousand one hundred seventy-nine |
---|

- 225829179 has 4 divisors, whose sum is
**250921320** - The reverse of 225829179 is
**971928522** - Previous prime number is
**9**

- Is Prime? no
- Number parity odd
- Number length 9
- Sum of Digits 45
- Digital Root 9

Name | nine hundred fifty million seven hundred thirty-seven thousand four hundred three |
---|

- 950737403 has 16 divisors, whose sum is
**1186099200** - The reverse of 950737403 is
**304737059** - Previous prime number is
**2111**

- Is Prime? no
- Number parity odd
- Number length 9
- Sum of Digits 38
- Digital Root 2

Name | one billion one hundred sixty-six million three hundred eighty-three thousand two hundred thirty-nine |
---|

- 1166383239 has 8 divisors, whose sum is
**1777345920** - The reverse of 1166383239 is
**9323836611** - Previous prime number is
**7**

- Is Prime? no
- Number parity odd
- Number length 10
- Sum of Digits 42
- Digital Root 6