Graph y=1-3cos(3(x-pi/3))

$y = 1 - 3 \cos (3 (x - \frac{π}{3}))$

Step 1

Rewrite the expression as $- 3 \cos (3 x - π) + 1$ .

$- 3 \cos (3 x - π) + 1$

Step 2

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

$a = - 3$

$b = 3$

$c = π$

$d = 1$

Step 3

Find the amplitude $| a |$ .

Amplitude: $3$

Step 4

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

Find the period of $- 3 \cos (3 x - π)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

Find the period of $1$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

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

$\frac{2 π}{3}$

Step 5

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

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

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

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

Phase Shift: $\frac{π}{3}$

Step 6

List the properties of the trigonometric function.

Amplitude: $3$

Period: $\frac{2 π}{3}$

Phase Shift: $\frac{π}{3}$ ( $\frac{π}{3}$ to the right)

Vertical Shift: $1$

Step 7

Select a few points to graph.

Find the point at $x = \frac{π}{3}$ .

Replace the variable $x$ with $\frac{π}{3}$ in the expression.

$f (\frac{π}{3}) = 1 - 3 \cos (3 (\frac{π}{3}) - π)$

Simplify the result.

Simplify each term.

Cancel the common factor of $3$ .

Cancel the common factor.

$f (\frac{π}{3}) = 1 - 3 \cos (3 (\frac{π}{3}) - π)$

Rewrite the expression.

$f (\frac{π}{3}) = 1 - 3 \cos (π - π)$

Subtract $π$ from $π$ .

$f (\frac{π}{3}) = 1 - 3 \cos (0)$

The exact value of $\cos (0)$ is $1$ .

$f (\frac{π}{3}) = 1 - 3 \cdot 1$

Multiply $- 3$ by $1$ .

$f (\frac{π}{3}) = 1 - 3$

Subtract $3$ from $1$ .

$f (\frac{π}{3}) = - 2$

The final answer is $- 2$ .

$- 2$

Find the point at $x = \frac{π}{2}$ .

Replace the variable $x$ with $\frac{π}{2}$ in the expression.

$f (\frac{π}{2}) = 1 - 3 \cos (3 (\frac{π}{2}) - π)$

Simplify the result.

Simplify each term.

Combine $3$ and $\frac{π}{2}$ .

$f (\frac{π}{2}) = 1 - 3 \cos (\frac{3 π}{2} - π)$

To write $- π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{π}{2}) = 1 - 3 \cos (\frac{3 π}{2} - π \cdot \frac{2}{2})$

Combine $- π$ and $\frac{2}{2}$ .

$f (\frac{π}{2}) = 1 - 3 \cos (\frac{3 π}{2} + \frac{- π \cdot 2}{2})$

Combine the numerators over the common denominator.

$f (\frac{π}{2}) = 1 - 3 \cos (\frac{3 π - π \cdot 2}{2})$

Simplify the numerator.

Multiply $2$ by $- 1$ .

$f (\frac{π}{2}) = 1 - 3 \cos (\frac{3 π - 2 π}{2})$

Subtract $2 π$ from $3 π$ .

$f (\frac{π}{2}) = 1 - 3 \cos (\frac{π}{2})$

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$f (\frac{π}{2}) = 1 - 3 \cdot 0$

Multiply $- 3$ by $0$ .

$f (\frac{π}{2}) = 1 + 0$

Add $1$ and $0$ .

$f (\frac{π}{2}) = 1$

The final answer is $1$ .

$1$

Find the point at $x = \frac{2 π}{3}$ .

Replace the variable $x$ with $\frac{2 π}{3}$ in the expression.

$f (\frac{2 π}{3}) = 1 - 3 \cos (3 (\frac{2 π}{3}) - π)$

Simplify the result.

Simplify each term.

Cancel the common factor of $3$ .

Cancel the common factor.

$f (\frac{2 π}{3}) = 1 - 3 \cos (3 (\frac{2 π}{3}) - π)$

Rewrite the expression.

$f (\frac{2 π}{3}) = 1 - 3 \cos (2 π - π)$

Subtract $π$ from $2 π$ .

$f (\frac{2 π}{3}) = 1 - 3 \cos (π)$

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.

$f (\frac{2 π}{3}) = 1 - 3 (- \cos (0))$

The exact value of $\cos (0)$ is $1$ .

$f (\frac{2 π}{3}) = 1 - 3 (- 1 \cdot 1)$

Multiply $- 3 (- 1 \cdot 1)$ .

Multiply $- 1$ by $1$ .

$f (\frac{2 π}{3}) = 1 - 3 \cdot - 1$

Multiply $- 3$ by $- 1$ .

$f (\frac{2 π}{3}) = 1 + 3$

Add $1$ and $3$ .

$f (\frac{2 π}{3}) = 4$

The final answer is $4$ .

$4$

Find the point at $x = \frac{5 π}{6}$ .

Replace the variable $x$ with $\frac{5 π}{6}$ in the expression.

$f (\frac{5 π}{6}) = 1 - 3 \cos (3 (\frac{5 π}{6}) - π)$

Simplify the result.

Simplify each term.

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

$f (\frac{5 π}{6}) = 1 - 3 \cos (3 (\frac{5 π}{3 (2)}) - π)$

Cancel the common factor.

$f (\frac{5 π}{6}) = 1 - 3 \cos (3 (\frac{5 π}{3 \cdot 2}) - π)$

Rewrite the expression.

$f (\frac{5 π}{6}) = 1 - 3 \cos (\frac{5 π}{2} - π)$

To write $- π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{5 π}{6}) = 1 - 3 \cos (\frac{5 π}{2} - π \cdot \frac{2}{2})$

Combine $- π$ and $\frac{2}{2}$ .

$f (\frac{5 π}{6}) = 1 - 3 \cos (\frac{5 π}{2} + \frac{- π \cdot 2}{2})$

Combine the numerators over the common denominator.

$f (\frac{5 π}{6}) = 1 - 3 \cos (\frac{5 π - π \cdot 2}{2})$

Simplify the numerator.

Multiply $2$ by $- 1$ .

$f (\frac{5 π}{6}) = 1 - 3 \cos (\frac{5 π - 2 π}{2})$

Subtract $2 π$ from $5 π$ .

$f (\frac{5 π}{6}) = 1 - 3 \cos (\frac{3 π}{2})$

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

$f (\frac{5 π}{6}) = 1 - 3 \cos (\frac{π}{2})$

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$f (\frac{5 π}{6}) = 1 - 3 \cdot 0$

Multiply $- 3$ by $0$ .

$f (\frac{5 π}{6}) = 1 + 0$

Add $1$ and $0$ .

$f (\frac{5 π}{6}) = 1$

The final answer is $1$ .

$1$

Find the point at $x = π$ .

Replace the variable $x$ with $π$ in the expression.

$f (π) = 1 - 3 \cos (3 (π) - π)$

Simplify the result.

Simplify each term.

Subtract $π$ from $3 π$ .

$f (π) = 1 - 3 \cos (2 π)$

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$f (π) = 1 - 3 \cos (0)$

The exact value of $\cos (0)$ is $1$ .

$f (π) = 1 - 3 \cdot 1$

Multiply $- 3$ by $1$ .

$f (π) = 1 - 3$

Subtract $3$ from $1$ .

$f (π) = - 2$

The final answer is $- 2$ .

$- 2$

List the points in a table.

$\begin{array}{c} x & f (x) \\ \frac{π}{3} & - 2 \\ \frac{π}{2} & 1 \\ \frac{2 π}{3} & 4 \\ \frac{5 π}{6} & 1 \\ π & - 2 \end{array}$

Step 8

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

Amplitude: $3$

Period: $\frac{2 π}{3}$

Phase Shift: $\frac{π}{3}$ ( $\frac{π}{3}$ to the right)

Vertical Shift: $1$

$\begin{array}{c} x & f (x) \\ \frac{π}{3} & - 2 \\ \frac{π}{2} & 1 \\ \frac{2 π}{3} & 4 \\ \frac{5 π}{6} & 1 \\ π & - 2 \end{array}$

Step 9

Graph y=1-3cos(3(x-pi/3))

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