Use the form <math><mstyle displaystyle="true"><mi>a</mi><mi>sin</mi><mrow><mo>(</mo><mi>b</mi><mi>x</mi><mo>-</mo><mi>c</mi><mo>)</mo></mrow><mo>+</mo><mi>d</mi></mstyle></math> to find the variables used to find the amplitude, period, phase shift, and vertical shift.

Find the amplitude <math><mstyle displaystyle="true"><mrow><mo>|</mo><mi>a</mi><mo>|</mo></mrow></mstyle></math> .

Amplitude: <math><mstyle displaystyle="true"><mn>4</mn></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"><mn>1</mn></mstyle></math> in the formula for period.

The absolute value is the distance between a number and zero. The distance between <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Divide <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

The phase shift of the function can be calculated from <math><mstyle displaystyle="true"><mfrac><mrow><mi>c</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> .

Phase Shift: <math><mstyle displaystyle="true"><mfrac><mrow><mi>c</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>

Replace the values of <math><mstyle displaystyle="true"><mi>c</mi></mstyle></math> and <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> in the equation for phase shift.

Phase Shift: <math><mstyle displaystyle="true"><mfrac><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></mfrac></mstyle></math>

Divide <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Phase Shift: <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math>

Phase Shift: <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math>

Find the vertical shift <math><mstyle displaystyle="true"><mi>d</mi></mstyle></math> .

Vertical Shift: <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math>

List the properties of the trigonometric function.

Amplitude: <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math>

Period: <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math>

Phase Shift: <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> (<math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> to the right)

Vertical Shift: <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math>

Find the point at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mn>0</mn></mstyle></math> .

Replace the variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> in the expression.

Simplify the result.

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

Multiply <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Find the point at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mfrac><mrow><mi>π</mi></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"><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> in the expression.

Simplify the result.

The exact value of <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

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

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

Find the point at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mi>π</mi></mstyle></math> .

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

Simplify the result.

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

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

Multiply <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Find the point at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi>π</mi></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"><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> in the expression.

Simplify the result.

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

The exact value of <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

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

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

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

Find the point at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mn>2</mn><mi>π</mi></mstyle></math> .

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

Simplify the result.

Subtract full rotations of <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> until the angle is greater than or equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and less than <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> .

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

Multiply <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

List the points in a table.

The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.

Amplitude: <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math>

Period: <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math>

Phase Shift: <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> (<math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> to the right)

Vertical Shift: <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math>

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Name | one billion one hundred twenty-four million seven hundred fifty-nine thousand three hundred sixty-three |
---|

- 1124759363 has 32 divisors, whose sum is
**1450939392** - The reverse of 1124759363 is
**3639574211** - Previous prime number is
**251**

- Is Prime? no
- Number parity odd
- Number length 10
- Sum of Digits 41
- Digital Root 5

Name | eight hundred twenty-four million nine hundred twenty-six thousand eighty-seven |
---|

- 824926087 has 4 divisors, whose sum is
**831421696** - The reverse of 824926087 is
**780629428** - Previous prime number is
**127**

- Is Prime? no
- Number parity odd
- Number length 9
- Sum of Digits 46
- Digital Root 1

Name | two billion one hundred thirty-nine million one hundred fourteen thousand forty-two |
---|

- 2139114042 has 32 divisors, whose sum is
**3661200000** - The reverse of 2139114042 is
**2404119312** - Previous prime number is
**2699**

- Is Prime? no
- Number parity even
- Number length 10
- Sum of Digits 27
- Digital Root 9