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>3</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> .

Period: <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>0.5</mn></mstyle></math> in the formula for period.

Period: <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mrow><mo>|</mo><mn>0.5</mn><mo>|</mo></mrow></mrow></mfrac></mstyle></math>

Solve the equation.

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>0.5</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>0.5</mn></mstyle></math> .

Period: <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mn>0.5</mn></mrow></mfrac></mstyle></math>

Replace <math><mstyle displaystyle="true"><mi>π</mi></mstyle></math> with an approximation.

Period: <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mo>⋅</mo><mn>3.14159265</mn></mrow><mrow><mn>0.5</mn></mrow></mfrac></mstyle></math>

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>3.14159265</mn></mstyle></math> .

Period: <math><mstyle displaystyle="true"><mfrac><mrow><mn>6.2831853</mn></mrow><mrow><mn>0.5</mn></mrow></mfrac></mstyle></math>

Divide <math><mstyle displaystyle="true"><mn>6.2831853</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>0.5</mn></mstyle></math> .

Period: <math><mstyle displaystyle="true"><mn>12.56637061</mn></mstyle></math>

Period: <math><mstyle displaystyle="true"><mn>12.56637061</mn></mstyle></math>

Period: <math><mstyle displaystyle="true"><mn>12.56637061</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><mn>1.57079632</mn></mrow><mrow><mn>0.5</mn></mrow></mfrac></mstyle></math>

Divide <math><mstyle displaystyle="true"><mo>-</mo><mn>1.57079632</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>0.5</mn></mstyle></math> .

Phase Shift: <math><mstyle displaystyle="true"><mo>-</mo><mn>3.14159265</mn></mstyle></math>

Phase Shift: <math><mstyle displaystyle="true"><mo>-</mo><mn>3.14159265</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>3</mn></mstyle></math>

Period: <math><mstyle displaystyle="true"><mn>12.56637061</mn></mstyle></math>

Phase Shift: <math><mstyle displaystyle="true"><mo>-</mo><mn>3.14159265</mn></mstyle></math> (<math><mstyle displaystyle="true"><mn>3.14159265</mn></mstyle></math> to the left)

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

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