Use the form <math><mstyle displaystyle="true"><mi>a</mi><mi>cos</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>1</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>2</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>2</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

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

Cancel the common factor.

Divide <math><mstyle displaystyle="true"><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><mo>-</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> .

Phase Shift: <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>2</mn><mo>⋅</mo><mn>4</mn></mrow></mfrac></mstyle></math>

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Phase Shift: <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>8</mn></mrow></mfrac></mstyle></math>

Phase Shift: <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>8</mn></mrow></mfrac></mstyle></math>

Phase Shift: <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>8</mn></mrow></mfrac></mstyle></math>

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

Vertical Shift: <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math>

List the properties of the trigonometric function.

Amplitude: <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math>

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

Phase Shift: <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>8</mn></mrow></mfrac></mstyle></math> (<math><mstyle displaystyle="true"><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>8</mn></mrow></mfrac></mstyle></math> to the left)

Vertical Shift: <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math>

Find the point at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>8</mn></mrow></mfrac></mstyle></math> .

Replace the variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>8</mn></mrow></mfrac></mstyle></math> in the expression.

Simplify the result.

Simplify each term.

Simplify each term.

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

Move the leading negative in <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>8</mn></mrow></mfrac></mstyle></math> into the numerator.

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>8</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Move the negative in front of the fraction.

Combine the numerators over the common denominator.

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>3</mn><mi>π</mi></mstyle></math> and <math><mstyle displaystyle="true"><mn>3</mn><mi>π</mi></mstyle></math> .

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

The exact value of <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Find the point at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>8</mn></mrow></mfrac></mstyle></math> .

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

Simplify the result.

Simplify each term.

Simplify each term.

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

Move the leading negative in <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>8</mn></mrow></mfrac></mstyle></math> into the numerator.

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>8</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Move the negative in front of the fraction.

Combine the numerators over the common denominator.

Add <math><mstyle displaystyle="true"><mo>-</mo><mi>π</mi></mstyle></math> and <math><mstyle displaystyle="true"><mn>3</mn><mi>π</mi></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

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

Cancel the common factors.

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

The exact value of <math><mstyle displaystyle="true"><mi>cos</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>0</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

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

Find the point at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>8</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>8</mn></mrow></mfrac></mstyle></math> in the expression.

Simplify the result.

Simplify each term.

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

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>8</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Combine the numerators over the common denominator.

Add <math><mstyle displaystyle="true"><mi>π</mi></mstyle></math> and <math><mstyle displaystyle="true"><mn>3</mn><mi>π</mi></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Cancel the common factor.

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

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

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

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>1</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>8</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>8</mn></mrow></mfrac></mstyle></math> in the expression.

Simplify the result.

Simplify each term.

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

Cancel the common factor.

Rewrite the expression.

Combine the numerators over the common denominator.

Add <math><mstyle displaystyle="true"><mn>3</mn><mi>π</mi></mstyle></math> and <math><mstyle displaystyle="true"><mn>3</mn><mi>π</mi></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>6</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>6</mn><mi>π</mi></mstyle></math> .

Cancel the common factors.

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

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>cos</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>0</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

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

Find the point at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><mrow><mn>8</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>5</mn><mi>π</mi></mrow><mrow><mn>8</mn></mrow></mfrac></mstyle></math> in the expression.

Simplify the result.

Simplify each term.

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

Cancel the common factor.

Rewrite the expression.

Combine the numerators over the common denominator.

Add <math><mstyle displaystyle="true"><mn>5</mn><mi>π</mi></mstyle></math> and <math><mstyle displaystyle="true"><mn>3</mn><mi>π</mi></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>8</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>8</mn><mi>π</mi></mstyle></math> .

Cancel the common factors.

Factor <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

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

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>cos</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><mn>2</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>1</mn></mstyle></math>

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

Phase Shift: <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>8</mn></mrow></mfrac></mstyle></math> (<math><mstyle displaystyle="true"><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>8</mn></mrow></mfrac></mstyle></math> to the left)

Vertical Shift: <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math>

Do you know how to Graph y=cos(2x+(3pi)/4)+1? If not, you can write to our math experts in our application. The best solution for your task you can find above on this page.

Name | one hundred million two hundred seventy-five thousand eight hundred seventeen |
---|

- 100275817 has 4 divisors, whose sum is
**100855620** - The reverse of 100275817 is
**718572001** - Previous prime number is
**173**

- Is Prime? no
- Number parity odd
- Number length 9
- Sum of Digits 31
- Digital Root 4

Name | one billion six hundred fourteen million eight hundred eighty-five thousand three hundred forty-four |
---|

- 1614885344 has 128 divisors, whose sum is
**12435771888** - The reverse of 1614885344 is
**4435884161** - Previous prime number is
**71**

- Is Prime? no
- Number parity even
- Number length 10
- Sum of Digits 44
- Digital Root 8

Name | four hundred twenty-three million three hundred nine thousand five hundred seventy-three |
---|

- 423309573 has 4 divisors, whose sum is
**470343980** - The reverse of 423309573 is
**375903324** - Previous prime number is
**9**

- Is Prime? no
- Number parity odd
- Number length 9
- Sum of Digits 36
- Digital Root 9