Graph y=4-3sin(2/5(x+1))

$y = 4 - 3 \sin (\frac{2}{5} (x + 1))$

Rewrite the expression as $- 3 \sin (\frac{2 x}{5} + \frac{2}{5}) + 4$ .

$- 3 \sin (\frac{2 x}{5} + \frac{2}{5}) + 4$

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

$a = - 3$

$b = \frac{2}{5}$

$c = - \frac{2}{5}$

$d = 4$

Find the amplitude $| a |$ .

Amplitude: $3$

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

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

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{2}{5}$ in the formula for period.

$\frac{2 π}{| \frac{2}{5} |}$

$\frac{2}{5}$ is approximately $0.4$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{5}{2}$

Cancel the common factor of $2$ .

Factor $2$ out of $2 π$ .

$2 (π) \frac{5}{2}$

Cancel the common factor.

$2 π \frac{5}{2}$

Rewrite the expression.

$π \cdot 5$

Move $5$ to the left of $π$ .

$5 π$

Find the period of $4$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{2}{5}$ in the formula for period.

$\frac{2 π}{| \frac{2}{5} |}$

$\frac{2}{5}$ is approximately $0.4$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{5}{2}$

Cancel the common factor of $2$ .

Factor $2$ out of $2 π$ .

$2 (π) \frac{5}{2}$

Cancel the common factor.

$2 π \frac{5}{2}$

Rewrite the expression.

$π \cdot 5$

Move $5$ to the left of $π$ .

$5 π$

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

$5 π$

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: $\frac{- \frac{2}{5}}{\frac{2}{5}}$

Cancel the common factor of $\frac{2}{5}$ .

Cancel the common factor.

Phase Shift: $\frac{- \frac{2}{5}}{\frac{2}{5}}$

Divide $- 1$ by $1$ .

Phase Shift: $- 1$

List the properties of the trigonometric function.

Amplitude: $3$

Period: $5 π$

Phase Shift: $- 1$ ( $1$ to the left)

Vertical Shift: $4$

Select a few points to graph.

Find the point at $x = - 1$ .

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

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

Simplify the result.

Simplify each term.

Combine the numerators over the common denominator.

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

Multiply $2$ by $- 1$ .

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

Add $- 2$ and $2$ .

$f (- 1) = 4 - 3 \sin (\frac{0}{5})$

Divide $0$ by $5$ .

$f (- 1) = 4 - 3 \sin (0)$

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

$f (- 1) = 4 - 3 \cdot 0$

Multiply $- 3$ by $0$ .

$f (- 1) = 4 + 0$

Add $4$ and $0$ .

$f (- 1) = 4$

The final answer is $4$ .

$4$

Find the point at $x = \frac{5 π}{4} - 1$ .

Replace the variable $x$ with $\frac{5 π}{4} - 1$ in the expression.

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

Simplify the result.

Simplify each term.

Combine the numerators over the common denominator.

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

Simplify each term.

Apply the distributive property.

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

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $2$ by $- 1$ .

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

Add $- 2$ and $2$ .

$f (\frac{5 π}{4} - 1) = 4 - 3 \sin (\frac{\frac{5 π}{2} + 0}{5})$

Add $\frac{5 π}{2}$ and $0$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $5$ .

Factor $5$ out of $5 π$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Multiply $- 3$ by $1$ .

$f (\frac{5 π}{4} - 1) = 4 - 3$

Subtract $3$ from $4$ .

$f (\frac{5 π}{4} - 1) = 1$

The final answer is $1$ .

$1$

Find the point at $x = \frac{5 π}{2} - 1$ .

Replace the variable $x$ with $\frac{5 π}{2} - 1$ in the expression.

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

Simplify the result.

Simplify each term.

Combine the numerators over the common denominator.

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

Simplify each term.

Apply the distributive property.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $2$ by $- 1$ .

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

Add $- 2$ and $2$ .

$f (\frac{5 π}{2} - 1) = 4 - 3 \sin (\frac{5 π + 0}{5})$

Add $5 π$ and $0$ .

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $π$ by $1$ .

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

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

$f (\frac{5 π}{2} - 1) = 4 - 3 \sin (0)$

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

$f (\frac{5 π}{2} - 1) = 4 - 3 \cdot 0$

Multiply $- 3$ by $0$ .

$f (\frac{5 π}{2} - 1) = 4 + 0$

Add $4$ and $0$ .

$f (\frac{5 π}{2} - 1) = 4$

The final answer is $4$ .

$4$

Find the point at $x = \frac{15 π}{4} - 1$ .

Replace the variable $x$ with $\frac{15 π}{4} - 1$ in the expression.

$f (\frac{15 π}{4} - 1) = 4 - 3 \sin (\frac{2 (\frac{15 π}{4} - 1)}{5} + \frac{2}{5})$

Simplify the result.

Simplify each term.

Combine the numerators over the common denominator.

$f (\frac{15 π}{4} - 1) = 4 - 3 \sin (\frac{2 (\frac{15 π}{4} - 1) + 2}{5})$

Simplify each term.

Apply the distributive property.

$f (\frac{15 π}{4} - 1) = 4 - 3 \sin (\frac{2 (\frac{15 π}{4}) + 2 \cdot - 1 + 2}{5})$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$f (\frac{15 π}{4} - 1) = 4 - 3 \sin (\frac{2 (\frac{15 π}{2 (2)}) + 2 \cdot - 1 + 2}{5})$

Cancel the common factor.

$f (\frac{15 π}{4} - 1) = 4 - 3 \sin (\frac{2 (\frac{15 π}{2 \cdot 2}) + 2 \cdot - 1 + 2}{5})$

Rewrite the expression.

$f (\frac{15 π}{4} - 1) = 4 - 3 \sin (\frac{\frac{15 π}{2} + 2 \cdot - 1 + 2}{5})$

Multiply $2$ by $- 1$ .

$f (\frac{15 π}{4} - 1) = 4 - 3 \sin (\frac{\frac{15 π}{2} - 2 + 2}{5})$

Add $- 2$ and $2$ .

$f (\frac{15 π}{4} - 1) = 4 - 3 \sin (\frac{\frac{15 π}{2} + 0}{5})$

Add $\frac{15 π}{2}$ and $0$ .

$f (\frac{15 π}{4} - 1) = 4 - 3 \sin (\frac{\frac{15 π}{2}}{5})$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{15 π}{4} - 1) = 4 - 3 \sin (\frac{15 π}{2} \cdot \frac{1}{5})$

Cancel the common factor of $5$ .

Factor $5$ out of $15 π$ .

$f (\frac{15 π}{4} - 1) = 4 - 3 \sin (\frac{5 (3 π)}{2} \cdot \frac{1}{5})$

Cancel the common factor.

$f (\frac{15 π}{4} - 1) = 4 - 3 \sin (\frac{5 (3 π)}{2} \cdot \frac{1}{5})$

Rewrite the expression.

$f (\frac{15 π}{4} - 1) = 4 - 3 \sin (\frac{3 π}{2})$

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.

$f (\frac{15 π}{4} - 1) = 4 - 3 (- \sin (\frac{π}{2}))$

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

$f (\frac{15 π}{4} - 1) = 4 - 3 (- 1 \cdot 1)$

Multiply $- 1$ by $1$ .

$f (\frac{15 π}{4} - 1) = 4 - 3 \cdot - 1$

Multiply $- 3$ by $- 1$ .

$f (\frac{15 π}{4} - 1) = 4 + 3$

Add $4$ and $3$ .

$f (\frac{15 π}{4} - 1) = 7$

The final answer is $7$ .

$7$

Find the point at $x = 5 π - 1$ .

Replace the variable $x$ with $5 π - 1$ in the expression.

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

Simplify the result.

Simplify each term.

Combine the numerators over the common denominator.

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

Simplify each term.

Apply the distributive property.

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

Multiply $5$ by $2$ .

$f (5 π - 1) = 4 - 3 \sin (\frac{10 π + 2 \cdot - 1 + 2}{5})$

Multiply $2$ by $- 1$ .

$f (5 π - 1) = 4 - 3 \sin (\frac{10 π - 2 + 2}{5})$

Add $- 2$ and $2$ .

$f (5 π - 1) = 4 - 3 \sin (\frac{10 π + 0}{5})$

Add $10 π$ and $0$ .

$f (5 π - 1) = 4 - 3 \sin (\frac{10 π}{5})$

Cancel the common factor of $10$ and $5$ .

Factor $5$ out of $10 π$ .

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

Cancel the common factors.

Factor $5$ out of $5$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2 π$ by $1$ .

$f (5 π - 1) = 4 - 3 \sin (2 π)$

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

$f (5 π - 1) = 4 - 3 \sin (0)$

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

$f (5 π - 1) = 4 - 3 \cdot 0$

Multiply $- 3$ by $0$ .

$f (5 π - 1) = 4 + 0$

Add $4$ and $0$ .

$f (5 π - 1) = 4$

The final answer is $4$ .

$4$

List the points in a table.

$\begin{array}{c} x & f (x) \\ - 1 & 4 \\ \frac{5 π}{4} - 1 & 1 \\ \frac{5 π}{2} - 1 & 4 \\ \frac{15 π}{4} - 1 & 7 \\ 5 π - 1 & 4 \end{array}$

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

Amplitude: $3$

Period: $5 π$

Phase Shift: $- 1$ ( $1$ to the left)

Vertical Shift: $4$

$\begin{array}{c} x & f (x) \\ - 1 & 4 \\ \frac{5 π}{4} - 1 & 1 \\ \frac{5 π}{2} - 1 & 4 \\ \frac{15 π}{4} - 1 & 7 \\ 5 π - 1 & 4 \end{array}$

Graph y=4-3sin(2/5(x+1))

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