To find the <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> coordinate of the vertex, set the inside of the absolute value <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mfrac><mrow><mi>π</mi><mi>x</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> . In this case, <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mfrac><mrow><mi>π</mi><mi>x</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mstyle></math> .

Solve the equation <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mfrac><mrow><mi>π</mi><mi>x</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mstyle></math> to find the <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> coordinate for the absolute value vertex.

Take the inverse cosine of both sides of the equation to extract <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> from inside the cosine.

The exact value of <math><mstyle displaystyle="true"><mi>arccos</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

Divide each term by <math><mstyle displaystyle="true"><mi>π</mi></mstyle></math> and simplify.

Divide each term in <math><mstyle displaystyle="true"><mi>π</mi><mi>x</mi><mo>=</mo><mi>π</mi></mstyle></math> by <math><mstyle displaystyle="true"><mi>π</mi></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mi>π</mi></mstyle></math> .

Cancel the common factor.

Divide <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mi>π</mi></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> to find the solution in the fourth quadrant.

Solve for <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

Multiply both sides of the equation by <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn></mrow><mrow><mi>π</mi></mrow></mfrac></mstyle></math> .

Simplify both sides of the equation.

Simplify <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn></mrow><mrow><mi>π</mi></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mi>π</mi><mi>x</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Cancel the common factor of <math><mstyle displaystyle="true"><mi>π</mi></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mi>π</mi></mstyle></math> out of <math><mstyle displaystyle="true"><mi>π</mi><mi>x</mi></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Simplify <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn></mrow><mrow><mi>π</mi></mrow></mfrac><mo>⋅</mo><mrow><mo>(</mo><mn>2</mn><mi>π</mi><mo>-</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> .

To write <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> as a fraction with a common denominator, multiply by <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Combine fractions.

Combine <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Combine the numerators over the common denominator.

Simplify the numerator.

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

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

Reduce the expression by cancelling the common factors.

Cancel the common factor of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Cancel the common factor of <math><mstyle displaystyle="true"><mi>π</mi></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mi>π</mi></mstyle></math> out of <math><mstyle displaystyle="true"><mn>3</mn><mi>π</mi></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Find the period of <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mfrac><mrow><mi>π</mi><mi>x</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> .

The period of the function can be calculated using <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mrow><mo>|</mo><mi>b</mi><mo>|</mo></mrow></mrow></mfrac></mstyle></math> .

Replace <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> with <math><mstyle displaystyle="true"><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> in the formula for period.

Multiply the numerator by the reciprocal of the denominator.

Cancel the common factor of <math><mstyle displaystyle="true"><mi>π</mi></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mi>π</mi></mstyle></math> out of <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

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

The period of the <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mfrac><mrow><mi>π</mi><mi>x</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> function is <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> so values will repeat every <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> radians in both directions.

Consolidate the answers.

Replace the variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mn>1</mn><mo>+</mo><mn>2</mn><mi>n</mi></mstyle></math> in the expression.

The absolute value vertex is <math><mstyle displaystyle="true"><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mn>2</mn><mi>n</mi><mo>,</mo><mn>2</mn><mrow><mo>|</mo><mi>cos</mi><mrow><mo>(</mo><mfrac><mrow><mi>π</mi><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mn>2</mn><mi>n</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><mo>|</mo></mrow><mo>)</mo></mrow></mstyle></math> .

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

Set-Builder Notation:

The absolute value can be graphed using the points around the vertex <math><mstyle displaystyle="true"><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mn>2</mn><mi>n</mi><mo>,</mo><mn>2</mn><mrow><mo>|</mo><mi>cos</mi><mrow><mo>(</mo><mfrac><mrow><mi>π</mi><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mn>2</mn><mi>n</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><mo>|</mo></mrow><mo>)</mo></mrow><mo>,</mo></mstyle></math>

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Name | eight hundred sixty-nine million seventy thousand twenty-five |
---|

- 869070025 has 16 divisors, whose sum is
**1257080832** - The reverse of 869070025 is
**520070968** - Previous prime number is
**223**

- Is Prime? no
- Number parity odd
- Number length 9
- Sum of Digits 37
- Digital Root 1

Name | one billion seven hundred twenty-seven million seven hundred eighty-six thousand six hundred eighty-one |
---|

- 1727786681 has 8 divisors, whose sum is
**1749535200** - The reverse of 1727786681 is
**1866877271** - Previous prime number is
**2393**

- Is Prime? no
- Number parity odd
- Number length 10
- Sum of Digits 53
- Digital Root 8

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

- 159105278 has 32 divisors, whose sum is
**267805440** - The reverse of 159105278 is
**872501951** - Previous prime number is
**163**

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