Algebra Examples

$y = 8 \cos (5 π x + \frac{3 π}{2}) - 9$

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.

$a = 8$

$b = 5 π$

$c = - \frac{3 π}{2}$

$d = - 9$

Step 2

Find the amplitude $| a |$ .

Amplitude: $8$

Step 3

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

Tap for more steps...

Step 3.1

Find the period of $8 \cos (5 π x + \frac{3 π}{2})$ .

Tap for more steps...

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 $5 π$ in the formula for period.

$\frac{2 π}{| 5 π |}$

Step 3.1.3

$5 π$ is approximately $15.70796326$ which is positive so remove the absolute value

$\frac{2 π}{5 π}$

Step 3.1.4

Cancel the common factor of $π$ .

Tap for more steps...

Step 3.1.4.1

Cancel the common factor.

$\frac{2 π}{5 π}$

Step 3.1.4.2

Rewrite the expression.

$\frac{2}{5}$

Step 3.2

Find the period of $- 9$ .

Tap for more steps...

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 $5 π$ in the formula for period.

$\frac{2 π}{| 5 π |}$

Step 3.2.3

$5 π$ is approximately $15.70796326$ which is positive so remove the absolute value

$\frac{2 π}{5 π}$

Step 3.2.4

Cancel the common factor of $π$ .

Tap for more steps...

Step 3.2.4.1

Cancel the common factor.

$\frac{2 π}{5 π}$

Step 3.2.4.2

Rewrite the expression.

$\frac{2}{5}$

Step 3.3

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

$\frac{2}{5}$

Step 4

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

Tap for more steps...

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: $\frac{- \frac{3 π}{2}}{5 π}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{3 π}{2} \cdot \frac{1}{5 π}$

Step 4.4

Cancel the common factor of $π$ .

Tap for more steps...

Step 4.4.1

Move the leading negative in $- \frac{3 π}{2}$ into the numerator.

Phase Shift: $\frac{- 3 π}{2} \cdot \frac{1}{5 π}$

Step 4.4.2

Factor $π$ out of $- 3 π$ .

Phase Shift: $\frac{π \cdot - 3}{2} \cdot \frac{1}{5 π}$

Step 4.4.3

Factor $π$ out of $5 π$ .

Phase Shift: $\frac{π \cdot - 3}{2} \cdot \frac{1}{π \cdot 5}$

Step 4.4.4

Cancel the common factor.

Phase Shift: $\frac{π \cdot - 3}{2} \cdot \frac{1}{π \cdot 5}$

Step 4.4.5

Rewrite the expression.

Phase Shift: $\frac{- 3}{2} \cdot \frac{1}{5}$

Step 4.5

Multiply $\frac{- 3}{2}$ by $\frac{1}{5}$ .

Phase Shift: $\frac{- 3}{2 \cdot 5}$

Step 4.6

Simplify the expression.

Tap for more steps...

Step 4.6.1

Multiply $2$ by $5$ .

Phase Shift: $\frac{- 3}{10}$

Step 4.6.2

Move the negative in front of the fraction.

Phase Shift: $- \frac{3}{10}$

Step 5

List the properties of the trigonometric function.

Amplitude: $8$

Period: $\frac{2}{5}$

Phase Shift: $- \frac{3}{10}$ ( $\frac{3}{10}$ to the left)

Vertical Shift: $- 9$

Step 6