Graph y=-3+4cos((5pi)/6(x+4))

$y = - 3 + 4 \cos (\frac{5 π}{6} (x + 4))$

Step 1

Rewrite the expression as $4 \cos (\frac{5 π x}{6} + \frac{10 π}{3}) - 3$ .

$4 \cos (\frac{5 π x}{6} + \frac{10 π}{3}) - 3$

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.

$a = 4$

$b = \frac{5 π}{6}$

$c = - \frac{10 π}{3}$

$d = - 3$

Step 3

Find the amplitude $| a |$ .

Amplitude: $4$

Step 4

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

Find the period of $4 \cos (\frac{5 π x}{6} + \frac{10 π}{3})$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{5 π}{6}$ in the formula for period.

$\frac{2 π}{| \frac{5 π}{6} |}$

$\frac{5 π}{6}$ is approximately $2.61799387$ which is positive so remove the absolute value

$\frac{2 π}{\frac{5 π}{6}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{6}{5 π}$

Cancel the common factor of $π$ .

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{6}{5 π}$

Factor $π$ out of $5 π$ .

$π \cdot 2 \frac{6}{π \cdot 5}$

Cancel the common factor.

$π \cdot 2 \frac{6}{π \cdot 5}$

Rewrite the expression.

$2 (\frac{6}{5})$

Combine $2$ and $\frac{6}{5}$ .

$\frac{2 \cdot 6}{5}$

Multiply $2$ by $6$ .

$\frac{12}{5}$

Find the period of $- 3$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{5 π}{6}$ in the formula for period.

$\frac{2 π}{| \frac{5 π}{6} |}$

$\frac{5 π}{6}$ is approximately $2.61799387$ which is positive so remove the absolute value

$\frac{2 π}{\frac{5 π}{6}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{6}{5 π}$

Cancel the common factor of $π$ .

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{6}{5 π}$

Factor $π$ out of $5 π$ .

$π \cdot 2 \frac{6}{π \cdot 5}$

Cancel the common factor.

$π \cdot 2 \frac{6}{π \cdot 5}$

Rewrite the expression.

$2 (\frac{6}{5})$

Combine $2$ and $\frac{6}{5}$ .

$\frac{2 \cdot 6}{5}$

Multiply $2$ by $6$ .

$\frac{12}{5}$

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

$\frac{12}{5}$

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: $\frac{- \frac{10 π}{3}}{\frac{5 π}{6}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{10 π}{3} \cdot \frac{6}{5 π}$

Cancel the common factor of $5 π$ .

Move the leading negative in $- \frac{10 π}{3}$ into the numerator.

Phase Shift: $\frac{- 10 π}{3} \cdot \frac{6}{5 π}$

Factor $5 π$ out of $- 10 π$ .

Phase Shift: $\frac{5 π (- 2)}{3} \cdot \frac{6}{5 π}$

Cancel the common factor.

Phase Shift: $\frac{5 π \cdot - 2}{3} \cdot \frac{6}{5 π}$

Rewrite the expression.

Phase Shift: $\frac{- 2}{3} \cdot 6$

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

Phase Shift: $- 2 \cdot 2$

Multiply $- 2$ by $2$ .

Phase Shift: $- 4$

Step 6

List the properties of the trigonometric function.

Amplitude: $4$

Period: $\frac{12}{5}$

Phase Shift: $- 4$ ( $4$ to the left)

Vertical Shift: $- 3$

Step 7

Select a few points to graph.

Find the point at $x = - 4$ .

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

$f (- 4) = - 3 + 4 \cos (\frac{5 π (- 4)}{6} + \frac{10 π}{3})$

Simplify the result.

Simplify each term.

Cancel the common factor of $- 4$ and $6$ .

Factor $2$ out of $5 π (- 4)$ .

$f (- 4) = - 3 + 4 \cos (\frac{2 (5 π \cdot (- 2))}{6} + \frac{10 π}{3})$

Cancel the common factors.

Factor $2$ out of $6$ .

$f (- 4) = - 3 + 4 \cos (\frac{2 (5 π \cdot (- 2))}{2 (3)} + \frac{10 π}{3})$

Cancel the common factor.

$f (- 4) = - 3 + 4 \cos (\frac{2 (5 π \cdot (- 2))}{2 \cdot 3} + \frac{10 π}{3})$

Rewrite the expression.

$f (- 4) = - 3 + 4 \cos (\frac{5 π \cdot (- 2)}{3} + \frac{10 π}{3})$

Multiply $- 2$ by $5$ .

$f (- 4) = - 3 + 4 \cos (\frac{- 10 π}{3} + \frac{10 π}{3})$

Move the negative in front of the fraction.

$f (- 4) = - 3 + 4 \cos (- \frac{10 π}{3} + \frac{10 π}{3})$

Combine the numerators over the common denominator.

$f (- 4) = - 3 + 4 \cos (\frac{- 10 π + 10 π}{3})$

Add $- 10 π$ and $10 π$ .

$f (- 4) = - 3 + 4 \cos (\frac{0}{3})$

Divide $0$ by $3$ .

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

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

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

Multiply $4$ by $1$ .

$f (- 4) = - 3 + 4$

Add $- 3$ and $4$ .

$f (- 4) = 1$

The final answer is $1$ .

$1$

Find the point at $x = - \frac{17}{5}$ .

Replace the variable $x$ with $- \frac{17}{5}$ in the expression.

$f (- \frac{17}{5}) = - 3 + 4 \cos (\frac{5 π (- \frac{17}{5})}{6} + \frac{10 π}{3})$

Simplify the result.

Simplify each term.

Simplify the numerator.

Multiply $- 1$ by $5$ .

$f (- \frac{17}{5}) = - 3 + 4 \cos (\frac{- 5 π \frac{17}{5}}{6} + \frac{10 π}{3})$

Combine $\frac{17}{5}$ and $- 5$ .

$f (- \frac{17}{5}) = - 3 + 4 \cos (\frac{\frac{17 \cdot - 5}{5} \cdot π}{6} + \frac{10 π}{3})$

Combine $\frac{17 \cdot - 5}{5}$ and $π$ .

$f (- \frac{17}{5}) = - 3 + 4 \cos (\frac{\frac{17 \cdot (- 5 π)}{5}}{6} + \frac{10 π}{3})$

Multiply $17$ by $- 5$ .

$f (- \frac{17}{5}) = - 3 + 4 \cos (\frac{\frac{- 85 π}{5}}{6} + \frac{10 π}{3})$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{- 85 π}{5}$ by cancelling the common factors.

Factor $5$ out of $- 85 π$ .

$f (- \frac{17}{5}) = - 3 + 4 \cos (\frac{\frac{5 (- 17 π)}{5}}{6} + \frac{10 π}{3})$

Factor $5$ out of $5$ .

$f (- \frac{17}{5}) = - 3 + 4 \cos (\frac{\frac{5 (- 17 π)}{5 (1)}}{6} + \frac{10 π}{3})$

Cancel the common factor.

$f (- \frac{17}{5}) = - 3 + 4 \cos (\frac{\frac{5 (- 17 π)}{5 \cdot 1}}{6} + \frac{10 π}{3})$

Rewrite the expression.

$f (- \frac{17}{5}) = - 3 + 4 \cos (\frac{\frac{- 17 π}{1}}{6} + \frac{10 π}{3})$

Divide $- 17 π$ by $1$ .

$f (- \frac{17}{5}) = - 3 + 4 \cos (\frac{- 17 π}{6} + \frac{10 π}{3})$

Move the negative in front of the fraction.

$f (- \frac{17}{5}) = - 3 + 4 \cos (- \frac{17 π}{6} + \frac{10 π}{3})$

To write $\frac{10 π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- \frac{17}{5}) = - 3 + 4 \cos (- \frac{17 π}{6} + \frac{10 π}{3} \cdot \frac{2}{2})$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{10 π}{3}$ by $\frac{2}{2}$ .

$f (- \frac{17}{5}) = - 3 + 4 \cos (- \frac{17 π}{6} + \frac{10 π \cdot 2}{3 \cdot 2})$

Multiply $3$ by $2$ .

$f (- \frac{17}{5}) = - 3 + 4 \cos (- \frac{17 π}{6} + \frac{10 π \cdot 2}{6})$

Combine the numerators over the common denominator.

$f (- \frac{17}{5}) = - 3 + 4 \cos (\frac{- 17 π + 10 π \cdot 2}{6})$

Simplify the numerator.

Multiply $2$ by $10$ .

$f (- \frac{17}{5}) = - 3 + 4 \cos (\frac{- 17 π + 20 π}{6})$

Add $- 17 π$ and $20 π$ .

$f (- \frac{17}{5}) = - 3 + 4 \cos (\frac{3 π}{6})$

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

Factor $3$ out of $3 π$ .

$f (- \frac{17}{5}) = - 3 + 4 \cos (\frac{3 (π)}{6})$

Cancel the common factors.

Factor $3$ out of $6$ .

$f (- \frac{17}{5}) = - 3 + 4 \cos (\frac{3 π}{3 \cdot 2})$

Cancel the common factor.

$f (- \frac{17}{5}) = - 3 + 4 \cos (\frac{3 π}{3 \cdot 2})$

Rewrite the expression.

$f (- \frac{17}{5}) = - 3 + 4 \cos (\frac{π}{2})$

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

$f (- \frac{17}{5}) = - 3 + 4 \cdot 0$

Multiply $4$ by $0$ .

$f (- \frac{17}{5}) = - 3 + 0$

Add $- 3$ and $0$ .

$f (- \frac{17}{5}) = - 3$

The final answer is $- 3$ .

$- 3$

Find the point at $x = - \frac{14}{5}$ .

Replace the variable $x$ with $- \frac{14}{5}$ in the expression.

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{5 π (- \frac{14}{5})}{6} + \frac{10 π}{3})$

Simplify the result.

Simplify each term.

Simplify the numerator.

Multiply $- 1$ by $5$ .

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{- 5 π \frac{14}{5}}{6} + \frac{10 π}{3})$

Combine $\frac{14}{5}$ and $- 5$ .

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{\frac{14 \cdot - 5}{5} \cdot π}{6} + \frac{10 π}{3})$

Combine $\frac{14 \cdot - 5}{5}$ and $π$ .

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{\frac{14 \cdot (- 5 π)}{5}}{6} + \frac{10 π}{3})$

Multiply $14$ by $- 5$ .

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{\frac{- 70 π}{5}}{6} + \frac{10 π}{3})$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{- 70 π}{5}$ by cancelling the common factors.

Factor $5$ out of $- 70 π$ .

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{\frac{5 (- 14 π)}{5}}{6} + \frac{10 π}{3})$

Factor $5$ out of $5$ .

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{\frac{5 (- 14 π)}{5 (1)}}{6} + \frac{10 π}{3})$

Cancel the common factor.

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{\frac{5 (- 14 π)}{5 \cdot 1}}{6} + \frac{10 π}{3})$

Rewrite the expression.

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{\frac{- 14 π}{1}}{6} + \frac{10 π}{3})$

Divide $- 14 π$ by $1$ .

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{- 14 π}{6} + \frac{10 π}{3})$

Cancel the common factor of $- 14$ and $6$ .

Factor $2$ out of $- 14 π$ .

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{2 (- 7 π)}{6} + \frac{10 π}{3})$

Cancel the common factors.

Factor $2$ out of $6$ .

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{2 (- 7 π)}{2 (3)} + \frac{10 π}{3})$

Cancel the common factor.

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{2 (- 7 π)}{2 \cdot 3} + \frac{10 π}{3})$

Rewrite the expression.

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{- 7 π}{3} + \frac{10 π}{3})$

Move the negative in front of the fraction.

$f (- \frac{14}{5}) = - 3 + 4 \cos (- \frac{7 π}{3} + \frac{10 π}{3})$

Combine the numerators over the common denominator.

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{- 7 π + 10 π}{3})$

Add $- 7 π$ and $10 π$ .

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{3 π}{3})$

Cancel the common factor of $3$ .

Cancel the common factor.

$f (- \frac{14}{5}) = - 3 + 4 \cos (\frac{3 π}{3})$

Divide $π$ by $1$ .

$f (- \frac{14}{5}) = - 3 + 4 \cos (π)$

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.

$f (- \frac{14}{5}) = - 3 + 4 (- \cos (0))$

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

$f (- \frac{14}{5}) = - 3 + 4 (- 1 \cdot 1)$

Multiply $4 (- 1 \cdot 1)$ .

Multiply $- 1$ by $1$ .

$f (- \frac{14}{5}) = - 3 + 4 \cdot - 1$

Multiply $4$ by $- 1$ .

$f (- \frac{14}{5}) = - 3 - 4$

Subtract $4$ from $- 3$ .

$f (- \frac{14}{5}) = - 7$

The final answer is $- 7$ .

$- 7$

Find the point at $x = - \frac{11}{5}$ .

Replace the variable $x$ with $- \frac{11}{5}$ in the expression.

$f (- \frac{11}{5}) = - 3 + 4 \cos (\frac{5 π (- \frac{11}{5})}{6} + \frac{10 π}{3})$

Simplify the result.

Simplify each term.

Simplify the numerator.

Multiply $- 1$ by $5$ .

$f (- \frac{11}{5}) = - 3 + 4 \cos (\frac{- 5 π \frac{11}{5}}{6} + \frac{10 π}{3})$

Combine $\frac{11}{5}$ and $- 5$ .

$f (- \frac{11}{5}) = - 3 + 4 \cos (\frac{\frac{11 \cdot - 5}{5} \cdot π}{6} + \frac{10 π}{3})$

Combine $\frac{11 \cdot - 5}{5}$ and $π$ .

$f (- \frac{11}{5}) = - 3 + 4 \cos (\frac{\frac{11 \cdot (- 5 π)}{5}}{6} + \frac{10 π}{3})$

Multiply $11$ by $- 5$ .

$f (- \frac{11}{5}) = - 3 + 4 \cos (\frac{\frac{- 55 π}{5}}{6} + \frac{10 π}{3})$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{- 55 π}{5}$ by cancelling the common factors.

Factor $5$ out of $- 55 π$ .

$f (- \frac{11}{5}) = - 3 + 4 \cos (\frac{\frac{5 (- 11 π)}{5}}{6} + \frac{10 π}{3})$

Factor $5$ out of $5$ .

$f (- \frac{11}{5}) = - 3 + 4 \cos (\frac{\frac{5 (- 11 π)}{5 (1)}}{6} + \frac{10 π}{3})$

Cancel the common factor.

$f (- \frac{11}{5}) = - 3 + 4 \cos (\frac{\frac{5 (- 11 π)}{5 \cdot 1}}{6} + \frac{10 π}{3})$

Rewrite the expression.

$f (- \frac{11}{5}) = - 3 + 4 \cos (\frac{\frac{- 11 π}{1}}{6} + \frac{10 π}{3})$

Divide $- 11 π$ by $1$ .

$f (- \frac{11}{5}) = - 3 + 4 \cos (\frac{- 11 π}{6} + \frac{10 π}{3})$

Move the negative in front of the fraction.

$f (- \frac{11}{5}) = - 3 + 4 \cos (- \frac{11 π}{6} + \frac{10 π}{3})$

To write $\frac{10 π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- \frac{11}{5}) = - 3 + 4 \cos (- \frac{11 π}{6} + \frac{10 π}{3} \cdot \frac{2}{2})$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{10 π}{3}$ by $\frac{2}{2}$ .

$f (- \frac{11}{5}) = - 3 + 4 \cos (- \frac{11 π}{6} + \frac{10 π \cdot 2}{3 \cdot 2})$

Multiply $3$ by $2$ .

$f (- \frac{11}{5}) = - 3 + 4 \cos (- \frac{11 π}{6} + \frac{10 π \cdot 2}{6})$

Combine the numerators over the common denominator.

$f (- \frac{11}{5}) = - 3 + 4 \cos (\frac{- 11 π + 10 π \cdot 2}{6})$

Simplify the numerator.

Multiply $2$ by $10$ .

$f (- \frac{11}{5}) = - 3 + 4 \cos (\frac{- 11 π + 20 π}{6})$

Add $- 11 π$ and $20 π$ .

$f (- \frac{11}{5}) = - 3 + 4 \cos (\frac{9 π}{6})$

Cancel the common factor of $9$ and $6$ .

Factor $3$ out of $9 π$ .

$f (- \frac{11}{5}) = - 3 + 4 \cos (\frac{3 (3 π)}{6})$

Cancel the common factors.

Factor $3$ out of $6$ .

$f (- \frac{11}{5}) = - 3 + 4 \cos (\frac{3 (3 π)}{3 (2)})$

Cancel the common factor.

$f (- \frac{11}{5}) = - 3 + 4 \cos (\frac{3 (3 π)}{3 \cdot 2})$

Rewrite the expression.

$f (- \frac{11}{5}) = - 3 + 4 \cos (\frac{3 π}{2})$

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

$f (- \frac{11}{5}) = - 3 + 4 \cos (\frac{π}{2})$

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

$f (- \frac{11}{5}) = - 3 + 4 \cdot 0$

Multiply $4$ by $0$ .

$f (- \frac{11}{5}) = - 3 + 0$

Add $- 3$ and $0$ .

$f (- \frac{11}{5}) = - 3$

The final answer is $- 3$ .

$- 3$

Find the point at $x = - \frac{8}{5}$ .

Replace the variable $x$ with $- \frac{8}{5}$ in the expression.

$f (- \frac{8}{5}) = - 3 + 4 \cos (\frac{5 π (- \frac{8}{5})}{6} + \frac{10 π}{3})$

Simplify the result.

Simplify each term.

Simplify the numerator.

Multiply $- 1$ by $5$ .

$f (- \frac{8}{5}) = - 3 + 4 \cos (\frac{- 5 π \frac{8}{5}}{6} + \frac{10 π}{3})$

Combine $\frac{8}{5}$ and $- 5$ .

$f (- \frac{8}{5}) = - 3 + 4 \cos (\frac{\frac{8 \cdot - 5}{5} \cdot π}{6} + \frac{10 π}{3})$

Combine $\frac{8 \cdot - 5}{5}$ and $π$ .

$f (- \frac{8}{5}) = - 3 + 4 \cos (\frac{\frac{8 \cdot (- 5 π)}{5}}{6} + \frac{10 π}{3})$

Multiply $8$ by $- 5$ .

$f (- \frac{8}{5}) = - 3 + 4 \cos (\frac{\frac{- 40 π}{5}}{6} + \frac{10 π}{3})$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{- 40 π}{5}$ by cancelling the common factors.

Factor $5$ out of $- 40 π$ .

$f (- \frac{8}{5}) = - 3 + 4 \cos (\frac{\frac{5 (- 8 π)}{5}}{6} + \frac{10 π}{3})$

Factor $5$ out of $5$ .

$f (- \frac{8}{5}) = - 3 + 4 \cos (\frac{\frac{5 (- 8 π)}{5 (1)}}{6} + \frac{10 π}{3})$

Cancel the common factor.

$f (- \frac{8}{5}) = - 3 + 4 \cos (\frac{\frac{5 (- 8 π)}{5 \cdot 1}}{6} + \frac{10 π}{3})$

Rewrite the expression.

$f (- \frac{8}{5}) = - 3 + 4 \cos (\frac{\frac{- 8 π}{1}}{6} + \frac{10 π}{3})$

Divide $- 8 π$ by $1$ .

$f (- \frac{8}{5}) = - 3 + 4 \cos (\frac{- 8 π}{6} + \frac{10 π}{3})$

Cancel the common factor of $- 8$ and $6$ .

Factor $2$ out of $- 8 π$ .

$f (- \frac{8}{5}) = - 3 + 4 \cos (\frac{2 (- 4 π)}{6} + \frac{10 π}{3})$

Cancel the common factors.

Factor $2$ out of $6$ .

$f (- \frac{8}{5}) = - 3 + 4 \cos (\frac{2 (- 4 π)}{2 (3)} + \frac{10 π}{3})$

Cancel the common factor.

$f (- \frac{8}{5}) = - 3 + 4 \cos (\frac{2 (- 4 π)}{2 \cdot 3} + \frac{10 π}{3})$

Rewrite the expression.

$f (- \frac{8}{5}) = - 3 + 4 \cos (\frac{- 4 π}{3} + \frac{10 π}{3})$

Move the negative in front of the fraction.

$f (- \frac{8}{5}) = - 3 + 4 \cos (- \frac{4 π}{3} + \frac{10 π}{3})$

Combine the numerators over the common denominator.

$f (- \frac{8}{5}) = - 3 + 4 \cos (\frac{- 4 π + 10 π}{3})$

Add $- 4 π$ and $10 π$ .

$f (- \frac{8}{5}) = - 3 + 4 \cos (\frac{6 π}{3})$

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

Factor $3$ out of $6 π$ .

$f (- \frac{8}{5}) = - 3 + 4 \cos (\frac{3 (2 π)}{3})$

Cancel the common factors.

Factor $3$ out of $3$ .

$f (- \frac{8}{5}) = - 3 + 4 \cos (\frac{3 (2 π)}{3 (1)})$

Cancel the common factor.

$f (- \frac{8}{5}) = - 3 + 4 \cos (\frac{3 (2 π)}{3 \cdot 1})$

Rewrite the expression.

$f (- \frac{8}{5}) = - 3 + 4 \cos (\frac{2 π}{1})$

Divide $2 π$ by $1$ .

$f (- \frac{8}{5}) = - 3 + 4 \cos (2 π)$

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

$f (- \frac{8}{5}) = - 3 + 4 \cos (0)$

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

$f (- \frac{8}{5}) = - 3 + 4 \cdot 1$

Multiply $4$ by $1$ .

$f (- \frac{8}{5}) = - 3 + 4$

Add $- 3$ and $4$ .

$f (- \frac{8}{5}) = 1$

The final answer is $1$ .

$1$

List the points in a table.

$\begin{array}{c} x & f (x) \\ - 4 & 1 \\ - \frac{17}{5} & - 3 \\ - \frac{14}{5} & - 7 \\ - \frac{11}{5} & - 3 \\ - \frac{8}{5} & 1 \end{array}$

Step 8

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

Amplitude: $4$

Period: $\frac{12}{5}$

Phase Shift: $- 4$ ( $4$ to the left)

Vertical Shift: $- 3$

$\begin{array}{c} x & f (x) \\ - 4 & 1 \\ - \frac{17}{5} & - 3 \\ - \frac{14}{5} & - 7 \\ - \frac{11}{5} & - 3 \\ - \frac{8}{5} & 1 \end{array}$

Step 9

Graph y=-3+4cos((5pi)/6(x+4))

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