Graph y=2-3cos(pi/3(x-1))

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

Step 1

Rewrite the expression as $- 3 \cos (\frac{π x}{3} - \frac{π}{3}) + 2$ .

$- 3 \cos (\frac{π x}{3} - \frac{π}{3}) + 2$

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.

$a = - 3$

$b = \frac{π}{3}$

$c = \frac{π}{3}$

$d = 2$

Step 3

Find the amplitude $| a |$ .

Amplitude: $3$

Step 4

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

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

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{π}{3}$ in the formula for period.

$\frac{2 π}{| \frac{π}{3} |}$

$\frac{π}{3}$ is approximately $1.04719755$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{π}$

Cancel the common factor of $π$ .

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{3}{π}$

Cancel the common factor.

$π \cdot 2 \frac{3}{π}$

Rewrite the expression.

$2 \cdot 3$

Multiply $2$ by $3$ .

$6$

Find the period of $2$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{π}{3}$ in the formula for period.

$\frac{2 π}{| \frac{π}{3} |}$

$\frac{π}{3}$ is approximately $1.04719755$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{π}$

Cancel the common factor of $π$ .

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{3}{π}$

Cancel the common factor.

$π \cdot 2 \frac{3}{π}$

Rewrite the expression.

$2 \cdot 3$

Multiply $2$ by $3$ .

$6$

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

$6$

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: $\frac{\frac{π}{3}}{\frac{π}{3}}$

Cancel the common factor of $\frac{π}{3}$ .

Cancel the common factor.

Phase Shift: $\frac{\frac{π}{3}}{\frac{π}{3}}$

Rewrite the expression.

Phase Shift: $1$

Step 6

List the properties of the trigonometric function.

Amplitude: $3$

Period: $6$

Phase Shift: $1$ ( $1$ to the right)

Vertical Shift: $2$

Step 7

Select a few points to graph.

Find the point at $x = 1$ .

Replace the variable $x$ with $1$ in the expression.

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

Simplify the result.

Simplify each term.

Combine the numerators over the common denominator.

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

Multiply $π$ by $1$ .

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

Subtract $π$ from $π$ .

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

Divide $0$ by $3$ .

$f (1) = 2 - 3 \cos (0)$

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

$f (1) = 2 - 3 \cdot 1$

Multiply $- 3$ by $1$ .

$f (1) = 2 - 3$

Subtract $3$ from $2$ .

$f (1) = - 1$

The final answer is $- 1$ .

$- 1$

Find the point at $x = \frac{5}{2}$ .

Replace the variable $x$ with $\frac{5}{2}$ in the expression.

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

Simplify the result.

Simplify each term.

Combine the numerators over the common denominator.

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

Simplify each term.

Combine $π$ and $\frac{5}{2}$ .

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

Move $5$ to the left of $π$ .

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

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

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

Combine $- π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $- 1$ .

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

Subtract $2 π$ from $5 π$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $3$ .

Factor $3$ out of $3 π$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

$f (\frac{5}{2}) = 2 - 3 \cdot 0$

Multiply $- 3$ by $0$ .

$f (\frac{5}{2}) = 2 + 0$

Add $2$ and $0$ .

$f (\frac{5}{2}) = 2$

The final answer is $2$ .

$2$

Find the point at $x = 4$ .

Replace the variable $x$ with $4$ in the expression.

$f (4) = 2 - 3 \cos (\frac{π (4)}{3} - \frac{π}{3})$

Simplify the result.

Simplify each term.

Combine the numerators over the common denominator.

$f (4) = 2 - 3 \cos (\frac{π (4) - π}{3})$

Move $4$ to the left of $π$ .

$f (4) = 2 - 3 \cos (\frac{4 π - π}{3})$

Subtract $π$ from $4 π$ .

$f (4) = 2 - 3 \cos (\frac{3 π}{3})$

Cancel the common factor of $3$ .

Cancel the common factor.

$f (4) = 2 - 3 \cos (\frac{3 π}{3})$

Divide $π$ by $1$ .

$f (4) = 2 - 3 \cos (π)$

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.

$f (4) = 2 - 3 (- \cos (0))$

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

$f (4) = 2 - 3 (- 1 \cdot 1)$

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

Multiply $- 1$ by $1$ .

$f (4) = 2 - 3 \cdot - 1$

Multiply $- 3$ by $- 1$ .

$f (4) = 2 + 3$

Add $2$ and $3$ .

$f (4) = 5$

The final answer is $5$ .

$5$

Find the point at $x = \frac{11}{2}$ .

Replace the variable $x$ with $\frac{11}{2}$ in the expression.

$f (\frac{11}{2}) = 2 - 3 \cos (\frac{π (\frac{11}{2})}{3} - \frac{π}{3})$

Simplify the result.

Simplify each term.

Combine the numerators over the common denominator.

$f (\frac{11}{2}) = 2 - 3 \cos (\frac{π (\frac{11}{2}) - π}{3})$

Simplify each term.

Combine $π$ and $\frac{11}{2}$ .

$f (\frac{11}{2}) = 2 - 3 \cos (\frac{\frac{π \cdot 11}{2} - π}{3})$

Move $11$ to the left of $π$ .

$f (\frac{11}{2}) = 2 - 3 \cos (\frac{\frac{11 π}{2} - π}{3})$

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

$f (\frac{11}{2}) = 2 - 3 \cos (\frac{\frac{11 π}{2} - π \cdot \frac{2}{2}}{3})$

Combine $- π$ and $\frac{2}{2}$ .

$f (\frac{11}{2}) = 2 - 3 \cos (\frac{\frac{11 π}{2} + \frac{- π \cdot 2}{2}}{3})$

Combine the numerators over the common denominator.

$f (\frac{11}{2}) = 2 - 3 \cos (\frac{\frac{11 π - π \cdot 2}{2}}{3})$

Simplify the numerator.

Multiply $2$ by $- 1$ .

$f (\frac{11}{2}) = 2 - 3 \cos (\frac{\frac{11 π - 2 π}{2}}{3})$

Subtract $2 π$ from $11 π$ .

$f (\frac{11}{2}) = 2 - 3 \cos (\frac{\frac{9 π}{2}}{3})$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{11}{2}) = 2 - 3 \cos (\frac{9 π}{2} \cdot \frac{1}{3})$

Cancel the common factor of $3$ .

Factor $3$ out of $9 π$ .

$f (\frac{11}{2}) = 2 - 3 \cos (\frac{3 (3 π)}{2} \cdot \frac{1}{3})$

Cancel the common factor.

$f (\frac{11}{2}) = 2 - 3 \cos (\frac{3 (3 π)}{2} \cdot \frac{1}{3})$

Rewrite the expression.

$f (\frac{11}{2}) = 2 - 3 \cos (\frac{3 π}{2})$

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

$f (\frac{11}{2}) = 2 - 3 \cos (\frac{π}{2})$

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

$f (\frac{11}{2}) = 2 - 3 \cdot 0$

Multiply $- 3$ by $0$ .

$f (\frac{11}{2}) = 2 + 0$

Add $2$ and $0$ .

$f (\frac{11}{2}) = 2$

The final answer is $2$ .

$2$

Find the point at $x = 7$ .

Replace the variable $x$ with $7$ in the expression.

$f (7) = 2 - 3 \cos (\frac{π (7)}{3} - \frac{π}{3})$

Simplify the result.

Simplify each term.

Combine the numerators over the common denominator.

$f (7) = 2 - 3 \cos (\frac{π (7) - π}{3})$

Move $7$ to the left of $π$ .

$f (7) = 2 - 3 \cos (\frac{7 π - π}{3})$

Subtract $π$ from $7 π$ .

$f (7) = 2 - 3 \cos (\frac{6 π}{3})$

Cancel the common factor of $6$ and $3$ .

Factor $3$ out of $6 π$ .

$f (7) = 2 - 3 \cos (\frac{3 (2 π)}{3})$

Cancel the common factors.

Factor $3$ out of $3$ .

$f (7) = 2 - 3 \cos (\frac{3 (2 π)}{3 (1)})$

Cancel the common factor.

$f (7) = 2 - 3 \cos (\frac{3 (2 π)}{3 \cdot 1})$

Rewrite the expression.

$f (7) = 2 - 3 \cos (\frac{2 π}{1})$

Divide $2 π$ by $1$ .

$f (7) = 2 - 3 \cos (2 π)$

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

$f (7) = 2 - 3 \cos (0)$

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

$f (7) = 2 - 3 \cdot 1$

Multiply $- 3$ by $1$ .

$f (7) = 2 - 3$

Subtract $3$ from $2$ .

$f (7) = - 1$

The final answer is $- 1$ .

$- 1$

List the points in a table.

$\begin{array}{c} x & f (x) \\ 1 & - 1 \\ \frac{5}{2} & 2 \\ 4 & 5 \\ \frac{11}{2} & 2 \\ 7 & - 1 \end{array}$

Step 8

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

Amplitude: $3$

Period: $6$

Phase Shift: $1$ ( $1$ to the right)

Vertical Shift: $2$

$\begin{array}{c} x & f (x) \\ 1 & - 1 \\ \frac{5}{2} & 2 \\ 4 & 5 \\ \frac{11}{2} & 2 \\ 7 & - 1 \end{array}$

Step 9

Graph y=2-3cos(pi/3(x-1))

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