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

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

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

$- 3 \cos (\frac{π x}{2} + π) + 2$

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

$a = - 3$

$b = \frac{π}{2}$

$c = - π$

$d = 2$

Find the amplitude $| a |$ .

Amplitude: $3$

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

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

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| \frac{π}{2} |}$

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

$\frac{2 π}{\frac{π}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{π}$

Cancel the common factor of $π$ .

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{2}{π}$

Cancel the common factor.

$π \cdot 2 \frac{2}{π}$

Rewrite the expression.

$2 \cdot 2$

Multiply $2$ by $2$ .

$4$

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

$\frac{2 π}{| \frac{π}{2} |}$

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

$\frac{2 π}{\frac{π}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{π}$

Cancel the common factor of $π$ .

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{2}{π}$

Cancel the common factor.

$π \cdot 2 \frac{2}{π}$

Rewrite the expression.

$2 \cdot 2$

Multiply $2$ by $2$ .

$4$

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

$4$

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: $\frac{- π}{\frac{π}{2}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- π \frac{2}{π}$

Cancel the common factor of $π$ .

Factor $π$ out of $- π$ .

Phase Shift: $π \cdot (- 1 \frac{2}{π})$

Cancel the common factor.

Phase Shift: $π \cdot (- 1 \frac{2}{π})$

Rewrite the expression.

Phase Shift: $- 1 \cdot 2$

Multiply $- 1$ by $2$ .

Phase Shift: $- 2$

List the properties of the trigonometric function.

Amplitude: $3$

Period: $4$

Phase Shift: $- 2$ ( $2$ to the left)

Vertical Shift: $2$

Select a few points to graph.

Find the point at $x = - 2$ .

Replace the variable $x$ with $- 2$ in the expression.

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

Simplify the result.

Simplify each term.

Cancel the common factor of $- 2$ and $2$ .

Factor $2$ out of $π (- 2)$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $π \cdot (- 1)$ by $1$ .

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

Move $- 1$ to the left of $π$ .

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

Rewrite $- 1 π$ as $- π$ .

$f (- 2) = 2 - 3 \cos (- π + π)$

Add $- π$ and $π$ .

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

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

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

Multiply $- 3$ by $1$ .

$f (- 2) = 2 - 3$

Subtract $3$ from $2$ .

$f (- 2) = - 1$

The final answer is $- 1$ .

$- 1$

Find the point at $x = - 1$ .

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

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

Simplify the result.

Simplify each term.

Move $- 1$ to the left of $π$ .

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

Move the negative in front of the fraction.

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

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

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

Combine $π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Move $2$ to the left of $π$ .

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

Add $- π$ and $2 π$ .

$f (- 1) = 2 - 3 \cos (\frac{π}{2})$

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

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

Multiply $- 3$ by $0$ .

$f (- 1) = 2 + 0$

Add $2$ and $0$ .

$f (- 1) = 2$

The final answer is $2$ .

$2$

Find the point at $x = 0$ .

Replace the variable $x$ with $0$ in the expression.

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

Simplify the result.

Simplify each term.

Cancel the common factor of $0$ and $2$ .

Factor $2$ out of $π (0)$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $π \cdot (0)$ by $1$ .

$f (0) = 2 - 3 \cos (π \cdot (0) + π)$

Multiply $π$ by $0$ .

$f (0) = 2 - 3 \cos (0 + π)$

Add $0$ and $π$ .

$f (0) = 2 - 3 \cos (π)$

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.

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

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

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

Multiply $- 1$ by $1$ .

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

Multiply $- 3$ by $- 1$ .

$f (0) = 2 + 3$

Add $2$ and $3$ .

$f (0) = 5$

The final answer is $5$ .

$5$

Find the point at $x = 1$ .

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

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

Simplify the result.

Simplify each term.

Multiply $π$ by $1$ .

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

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

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

Combine $π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Move $2$ to the left of $π$ .

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

Add $π$ and $2 π$ .

$f (1) = 2 - 3 \cos (\frac{3 π}{2})$

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

$f (1) = 2 - 3 \cos (\frac{π}{2})$

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

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

Multiply $- 3$ by $0$ .

$f (1) = 2 + 0$

Add $2$ and $0$ .

$f (1) = 2$

The final answer is $2$ .

$2$

Find the point at $x = 2$ .

Replace the variable $x$ with $2$ in the expression.

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

Simplify the result.

Simplify each term.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $π$ by $1$ .

$f (2) = 2 - 3 \cos (π + π)$

Add $π$ and $π$ .

$f (2) = 2 - 3 \cos (2 π)$

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

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

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

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

Multiply $- 3$ by $1$ .

$f (2) = 2 - 3$

Subtract $3$ from $2$ .

$f (2) = - 1$

The final answer is $- 1$ .

$- 1$

List the points in a table.

$\begin{array}{c} x & f (x) \\ - 2 & - 1 \\ - 1 & 2 \\ 0 & 5 \\ 1 & 2 \\ 2 & - 1 \end{array}$

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

Amplitude: $3$

Period: $4$

Phase Shift: $- 2$ ( $2$ to the left)

Vertical Shift: $2$

$\begin{array}{c} x & f (x) \\ - 2 & - 1 \\ - 1 & 2 \\ 0 & 5 \\ 1 & 2 \\ 2 & - 1 \end{array}$

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

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