Algebra Examples

$- 9 \cos (\frac{π}{2} x - 6) + 8$

Step 1

Use the form $a \cos (b x - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

$a = - 9$

$b = \frac{π}{2}$

$c = 6$

$d = 8$

Step 2

Find the amplitude $| a |$ .

Amplitude: $9$

Step 3

Find the period using the formula $\frac{2 π}{| b |}$ .

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Step 3.1

Find the period of $- 9 \cos (\frac{π x}{2} - 6)$ .

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Step 3.1.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.1.2

Replace $b$ with $\frac{π}{2}$ in the formula for period.

$\frac{2 π}{| \frac{π}{2} |}$

Step 3.1.3

$\frac{π}{2}$ is approximately $1.57079632$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{2}}$

Step 3.1.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{π}$

Step 3.1.5

Cancel the common factor of $π$ .

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Step 3.1.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{2}{π}$

Step 3.1.5.2

Cancel the common factor.

$π \cdot 2 \frac{2}{π}$

Step 3.1.5.3

Rewrite the expression.

$2 \cdot 2$

Step 3.1.6

Multiply $2$ by $2$ .

$4$

Step 3.2

Find the period of $8$ .

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Step 3.2.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.2.2

Replace $b$ with $\frac{π}{2}$ in the formula for period.

$\frac{2 π}{| \frac{π}{2} |}$

Step 3.2.3

$\frac{π}{2}$ is approximately $1.57079632$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{2}}$

Step 3.2.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{π}$

Step 3.2.5

Cancel the common factor of $π$ .

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Step 3.2.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{2}{π}$

Step 3.2.5.2

Cancel the common factor.

$π \cdot 2 \frac{2}{π}$

Step 3.2.5.3

Rewrite the expression.

$2 \cdot 2$

Step 3.2.6

Multiply $2$ by $2$ .

$4$

Step 3.3

The period of addition/subtraction of trig functions is the maximum of the individual periods.

$4$

Step 4

Find the phase shift using the formula $\frac{c}{b}$ .

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Step 4.1

The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Step 4.2

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{6}{\frac{π}{2}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $6 (\frac{2}{π})$

Step 4.4

Multiply $6 \frac{2}{π}$ .

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Step 4.4.1

Combine $6$ and $\frac{2}{π}$ .

Phase Shift: $\frac{6 \cdot 2}{π}$

Step 4.4.2

Multiply $6$ by $2$ .

Phase Shift: $\frac{12}{π}$

Step 5

List the properties of the trigonometric function.

Amplitude: $9$

Period: $4$

Phase Shift: $\frac{12}{π}$ ( $\frac{12}{π}$ to the right)

Vertical Shift: $8$

Step 6