Graph y=1/2sin(2theta+(5pi)/3)+2

$y = \frac{1}{2} \sin (2 θ + \frac{5 π}{3}) + 2$

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

$a = \frac{1}{2}$

$b = 2$

$c = - \frac{5 π}{3}$

$d = 2$

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{2}$

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

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

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

$\frac{2 π}{| b |}$

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 π}{2}$

Divide $π$ by $1$ .

$π$

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 $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 π}{2}$

Divide $π$ by $1$ .

$π$

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

$π$

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: $\frac{- \frac{5 π}{3}}{2}$

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

Multiply $2$ by $3$ .

Phase Shift: $- \frac{5 π}{6}$

List the properties of the trigonometric function.

Amplitude: $\frac{1}{2}$

Period: $π$

Phase Shift: $- \frac{5 π}{6}$ ( $\frac{5 π}{6}$ to the left)

Vertical Shift: $2$

Select a few points to graph.

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

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

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

Simplify the result.

Combine $2 (- \frac{5 π}{6})$ and $\frac{5 π}{3}$ using a common denominator.

To write $2 \cdot - 1 \frac{5 π}{6}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Combine $2 \cdot - 1 \frac{5 π}{6}$ and $\frac{3}{3}$ .

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

Combine the numerators over the common denominator.

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

Multiply $2$ by $- 1$ .

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

Simplify each term.

Simplify the numerator.

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

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

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{5 π}{3}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

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

Add $- 5 π$ and $5 π$ .

$f (- \frac{5 π}{6}) = \frac{\sin (\frac{0}{3})}{2} + 2$

Simplify the numerator.

Divide $0$ by $3$ .

$f (- \frac{5 π}{6}) = \frac{\sin (0)}{2} + 2$

The exact value of $\sin (0)$ is $0$ .

$f (- \frac{5 π}{6}) = \frac{0}{2} + 2$

Divide $0$ by $2$ .

$f (- \frac{5 π}{6}) = 0 + 2$

Add $0$ and $2$ .

$f (- \frac{5 π}{6}) = 2$

The final answer is $2$ .

$2$

Find the point at $x = - \frac{7 π}{12}$ .

Replace the variable $x$ with $- \frac{7 π}{12}$ in the expression.

$f (- \frac{7 π}{12}) = \frac{\sin (2 (- \frac{7 π}{12}) + \frac{5 π}{3})}{2} + 2$

Simplify the result.

Combine $2 (- \frac{7 π}{12})$ and $\frac{5 π}{3}$ using a common denominator.

To write $2 \cdot - 1 \frac{7 π}{12}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- \frac{7 π}{12}) = \frac{\sin (2 \cdot (- 1 \frac{7 π}{12}) \cdot \frac{3}{3} + \frac{5 π}{3})}{2} + 2$

Combine $2 \cdot - 1 \frac{7 π}{12}$ and $\frac{3}{3}$ .

$f (- \frac{7 π}{12}) = \frac{\sin (\frac{2 \cdot (- 1 \frac{7 π}{12}) \cdot 3}{3} + \frac{5 π}{3})}{2} + 2$

Combine the numerators over the common denominator.

$f (- \frac{7 π}{12}) = \frac{\sin (\frac{2 \cdot (- 1 \frac{7 π}{12}) \cdot 3 + 5 π}{3})}{2} + 2$

Multiply $2$ by $- 1$ .

$f (- \frac{7 π}{12}) = \frac{\sin (\frac{- 2 \frac{7 π}{12} \cdot 3 + 5 π}{3})}{2} + 2$

Simplify each term.

Simplify the numerator.

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

$f (- \frac{7 π}{12}) = \frac{\sin (\frac{2 (- 1) (\frac{7 π}{12}) \cdot 3 + 5 π}{3})}{2} + 2$

Factor $2$ out of $12$ .

$f (- \frac{7 π}{12}) = \frac{\sin (\frac{2 \cdot (- 1 \frac{7 π}{2 \cdot 6}) \cdot 3 + 5 π}{3})}{2} + 2$

Cancel the common factor.

$f (- \frac{7 π}{12}) = \frac{\sin (\frac{2 \cdot (- 1 \frac{7 π}{2 \cdot 6}) \cdot 3 + 5 π}{3})}{2} + 2$

Rewrite the expression.

$f (- \frac{7 π}{12}) = \frac{\sin (\frac{- \frac{7 π}{6} \cdot 3 + 5 π}{3})}{2} + 2$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{7 π}{6}$ into the numerator.

$f (- \frac{7 π}{12}) = \frac{\sin (\frac{\frac{- 7 π}{6} \cdot 3 + 5 π}{3})}{2} + 2$

Factor $3$ out of $6$ .

$f (- \frac{7 π}{12}) = \frac{\sin (\frac{\frac{- 7 π}{3 (2)} \cdot 3 + 5 π}{3})}{2} + 2$

Cancel the common factor.

$f (- \frac{7 π}{12}) = \frac{\sin (\frac{\frac{- 7 π}{3 \cdot 2} \cdot 3 + 5 π}{3})}{2} + 2$

Rewrite the expression.

$f (- \frac{7 π}{12}) = \frac{\sin (\frac{\frac{- 7 π}{2} + 5 π}{3})}{2} + 2$

Move the negative in front of the fraction.

$f (- \frac{7 π}{12}) = \frac{\sin (\frac{- \frac{(7) π}{2} + 5 π}{3})}{2} + 2$

To write $5 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- \frac{7 π}{12}) = \frac{\sin (\frac{- \frac{7 π}{2} + 5 π \cdot \frac{2}{2}}{3})}{2} + 2$

Combine $5 π$ and $\frac{2}{2}$ .

$f (- \frac{7 π}{12}) = \frac{\sin (\frac{- \frac{7 π}{2} + \frac{5 π \cdot 2}{2}}{3})}{2} + 2$

Combine the numerators over the common denominator.

$f (- \frac{7 π}{12}) = \frac{\sin (\frac{\frac{- 7 π + 5 π \cdot 2}{2}}{3})}{2} + 2$

Rewrite $\frac{- 7 π + 5 π \cdot 2}{2}$ in a factored form.

Multiply $2$ by $5$ .

$f (- \frac{7 π}{12}) = \frac{\sin (\frac{\frac{- 7 π + 10 π}{2}}{3})}{2} + 2$

Add $- 7 π$ and $10 π$ .

$f (- \frac{7 π}{12}) = \frac{\sin (\frac{\frac{3 π}{2}}{3})}{2} + 2$

Simplify the numerator.

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{7 π}{12}) = \frac{\sin (\frac{3 π}{2} \cdot \frac{1}{3})}{2} + 2$

Cancel the common factor of $3$ .

Factor $3$ out of $3 π$ .

$f (- \frac{7 π}{12}) = \frac{\sin (\frac{3 (π)}{2} \cdot \frac{1}{3})}{2} + 2$

Cancel the common factor.

$f (- \frac{7 π}{12}) = \frac{\sin (\frac{3 π}{2} \cdot \frac{1}{3})}{2} + 2$

Rewrite the expression.

$f (- \frac{7 π}{12}) = \frac{\sin (\frac{π}{2})}{2} + 2$

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$f (- \frac{7 π}{12}) = \frac{1}{2} + 2$

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

$f (- \frac{7 π}{12}) = \frac{1}{2} + 2 \cdot \frac{2}{2}$

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

$f (- \frac{7 π}{12}) = \frac{1}{2} + \frac{2 \cdot 2}{2}$

Combine the numerators over the common denominator.

$f (- \frac{7 π}{12}) = \frac{1 + 2 \cdot 2}{2}$

Simplify the numerator.

Multiply $2$ by $2$ .

$f (- \frac{7 π}{12}) = \frac{1 + 4}{2}$

Add $1$ and $4$ .

$f (- \frac{7 π}{12}) = \frac{5}{2}$

The final answer is $\frac{5}{2}$ .

$\frac{5}{2}$

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

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

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

Simplify the result.

Combine $2 (- \frac{π}{3})$ and $\frac{5 π}{3}$ using a common denominator.

To write $2 \cdot - 1 \frac{π}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Combine $2 \cdot - 1 \frac{π}{3}$ and $\frac{3}{3}$ .

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

Combine the numerators over the common denominator.

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

Multiply $2$ by $- 1$ .

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

Simplify each term.

Simplify the numerator.

Combine $- 2$ and $\frac{π}{3}$ .

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Add $- 2 π$ and $5 π$ .

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

Simplify the numerator.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $π$ by $1$ .

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

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

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

The exact value of $\sin (0)$ is $0$ .

$f (- \frac{π}{3}) = \frac{0}{2} + 2$

Divide $0$ by $2$ .

$f (- \frac{π}{3}) = 0 + 2$

Add $0$ and $2$ .

$f (- \frac{π}{3}) = 2$

The final answer is $2$ .

$2$

Find the point at $x = - \frac{π}{12}$ .

Replace the variable $x$ with $- \frac{π}{12}$ in the expression.

$f (- \frac{π}{12}) = \frac{\sin (2 (- \frac{π}{12}) + \frac{5 π}{3})}{2} + 2$

Simplify the result.

Combine $2 (- \frac{π}{12})$ and $\frac{5 π}{3}$ using a common denominator.

To write $2 \cdot - 1 \frac{π}{12}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- \frac{π}{12}) = \frac{\sin (2 \cdot (- 1 \frac{π}{12}) \cdot \frac{3}{3} + \frac{5 π}{3})}{2} + 2$

Combine $2 \cdot - 1 \frac{π}{12}$ and $\frac{3}{3}$ .

$f (- \frac{π}{12}) = \frac{\sin (\frac{2 \cdot (- 1 \frac{π}{12}) \cdot 3}{3} + \frac{5 π}{3})}{2} + 2$

Combine the numerators over the common denominator.

$f (- \frac{π}{12}) = \frac{\sin (\frac{2 \cdot (- 1 \frac{π}{12}) \cdot 3 + 5 π}{3})}{2} + 2$

Multiply $2$ by $- 1$ .

$f (- \frac{π}{12}) = \frac{\sin (\frac{- 2 \frac{π}{12} \cdot 3 + 5 π}{3})}{2} + 2$

Simplify each term.

Simplify the numerator.

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

$f (- \frac{π}{12}) = \frac{\sin (\frac{2 (- 1) (\frac{π}{12}) \cdot 3 + 5 π}{3})}{2} + 2$

Factor $2$ out of $12$ .

$f (- \frac{π}{12}) = \frac{\sin (\frac{2 \cdot (- 1 \frac{π}{2 \cdot 6}) \cdot 3 + 5 π}{3})}{2} + 2$

Cancel the common factor.

$f (- \frac{π}{12}) = \frac{\sin (\frac{2 \cdot (- 1 \frac{π}{2 \cdot 6}) \cdot 3 + 5 π}{3})}{2} + 2$

Rewrite the expression.

$f (- \frac{π}{12}) = \frac{\sin (\frac{- \frac{π}{6} \cdot 3 + 5 π}{3})}{2} + 2$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{π}{6}$ into the numerator.

$f (- \frac{π}{12}) = \frac{\sin (\frac{\frac{- π}{6} \cdot 3 + 5 π}{3})}{2} + 2$

Factor $3$ out of $6$ .

$f (- \frac{π}{12}) = \frac{\sin (\frac{\frac{- π}{3 (2)} \cdot 3 + 5 π}{3})}{2} + 2$

Cancel the common factor.

$f (- \frac{π}{12}) = \frac{\sin (\frac{\frac{- π}{3 \cdot 2} \cdot 3 + 5 π}{3})}{2} + 2$

Rewrite the expression.

$f (- \frac{π}{12}) = \frac{\sin (\frac{\frac{- π}{2} + 5 π}{3})}{2} + 2$

Move the negative in front of the fraction.

$f (- \frac{π}{12}) = \frac{\sin (\frac{- \frac{π}{2} + 5 π}{3})}{2} + 2$

To write $5 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- \frac{π}{12}) = \frac{\sin (\frac{- \frac{π}{2} + 5 π \cdot \frac{2}{2}}{3})}{2} + 2$

Combine $5 π$ and $\frac{2}{2}$ .

$f (- \frac{π}{12}) = \frac{\sin (\frac{- \frac{π}{2} + \frac{5 π \cdot 2}{2}}{3})}{2} + 2$

Combine the numerators over the common denominator.

$f (- \frac{π}{12}) = \frac{\sin (\frac{\frac{- π + 5 π \cdot 2}{2}}{3})}{2} + 2$

Rewrite $\frac{- π + 5 π \cdot 2}{2}$ in a factored form.

Multiply $2$ by $5$ .

$f (- \frac{π}{12}) = \frac{\sin (\frac{\frac{- π + 10 π}{2}}{3})}{2} + 2$

Add $- π$ and $10 π$ .

$f (- \frac{π}{12}) = \frac{\sin (\frac{\frac{9 π}{2}}{3})}{2} + 2$

Simplify the numerator.

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{π}{12}) = \frac{\sin (\frac{9 π}{2} \cdot \frac{1}{3})}{2} + 2$

Cancel the common factor of $3$ .

Factor $3$ out of $9 π$ .

$f (- \frac{π}{12}) = \frac{\sin (\frac{3 (3 π)}{2} \cdot \frac{1}{3})}{2} + 2$

Cancel the common factor.

$f (- \frac{π}{12}) = \frac{\sin (\frac{3 (3 π)}{2} \cdot \frac{1}{3})}{2} + 2$

Rewrite the expression.

$f (- \frac{π}{12}) = \frac{\sin (\frac{3 π}{2})}{2} + 2$

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$f (- \frac{π}{12}) = \frac{- \sin (\frac{π}{2})}{2} + 2$

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$f (- \frac{π}{12}) = \frac{- 1 \cdot 1}{2} + 2$

Multiply $- 1$ by $1$ .

$f (- \frac{π}{12}) = \frac{- 1}{2} + 2$

Move the negative in front of the fraction.

$f (- \frac{π}{12}) = - \frac{1}{2} + 2$

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

$f (- \frac{π}{12}) = - \frac{1}{2} + 2 \cdot \frac{2}{2}$

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

$f (- \frac{π}{12}) = - \frac{1}{2} + \frac{2 \cdot 2}{2}$

Combine the numerators over the common denominator.

$f (- \frac{π}{12}) = \frac{- 1 + 2 \cdot 2}{2}$

Simplify the numerator.

Multiply $2$ by $2$ .

$f (- \frac{π}{12}) = \frac{- 1 + 4}{2}$

Add $- 1$ and $4$ .

$f (- \frac{π}{12}) = \frac{3}{2}$

The final answer is $\frac{3}{2}$ .

$\frac{3}{2}$

Find the point at $x = \frac{π}{6}$ .

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

$f (\frac{π}{6}) = \frac{\sin (2 (\frac{π}{6}) + \frac{5 π}{3})}{2} + 2$

Simplify the result.

To write $2 \frac{π}{6}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Combine fractions.

Combine $2 \frac{π}{6}$ and $\frac{3}{3}$ .

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

Combine the numerators over the common denominator.

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

Simplify each term.

Simplify the numerator.

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

$f (\frac{π}{6}) = \frac{\sin (\frac{π + 5 π}{3})}{2} + 2$

Add $π$ and $5 π$ .

$f (\frac{π}{6}) = \frac{\sin (\frac{6 π}{3})}{2} + 2$

Simplify the numerator.

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

Factor $3$ out of $6 π$ .

$f (\frac{π}{6}) = \frac{\sin (\frac{3 (2 π)}{3})}{2} + 2$

Cancel the common factors.

Factor $3$ out of $3$ .

$f (\frac{π}{6}) = \frac{\sin (\frac{3 (2 π)}{3 (1)})}{2} + 2$

Cancel the common factor.

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

Rewrite the expression.

$f (\frac{π}{6}) = \frac{\sin (\frac{2 π}{1})}{2} + 2$

Divide $2 π$ by $1$ .

$f (\frac{π}{6}) = \frac{\sin (2 π)}{2} + 2$

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

$f (\frac{π}{6}) = \frac{\sin (0)}{2} + 2$

The exact value of $\sin (0)$ is $0$ .

$f (\frac{π}{6}) = \frac{0}{2} + 2$

Divide $0$ by $2$ .

$f (\frac{π}{6}) = 0 + 2$

Add $0$ and $2$ .

$f (\frac{π}{6}) = 2$

The final answer is $2$ .

$2$

List the points in a table.

$\begin{array}{c} x & f (x) \\ - \frac{5 π}{6} & 2 \\ - \frac{7 π}{12} & \frac{5}{2} \\ - \frac{π}{3} & 2 \\ - \frac{π}{12} & \frac{3}{2} \\ \frac{π}{6} & 2 \end{array}$

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

Amplitude: $\frac{1}{2}$

Period: $π$

Phase Shift: $- \frac{5 π}{6}$ ( $\frac{5 π}{6}$ to the left)

Vertical Shift: $2$

$\begin{array}{c} x & f (x) \\ - \frac{5 π}{6} & 2 \\ - \frac{7 π}{12} & \frac{5}{2} \\ - \frac{π}{3} & 2 \\ - \frac{π}{12} & \frac{3}{2} \\ \frac{π}{6} & 2 \end{array}$

Graph y=1/2sin(2theta+(5pi)/3)+2

Related Questions

Support LCMGCF.com by disabling your adblocker.