Graph f(x)=5sin(pi/2x-pi)+3

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

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

$a = 5$

$b = \frac{π}{2}$

$c = π$

$d = 3$

Find the amplitude $| a |$ .

Amplitude: $5$

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

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

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 $3$ .

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 $π$ .

Cancel the common factor.

Phase Shift: $π (\frac{2}{π})$

Rewrite the expression.

Phase Shift: $2$

List the properties of the trigonometric function.

Amplitude: $5$

Period: $4$

Phase Shift: $2$ ( $2$ to the right)

Vertical Shift: $3$

Select a few points to graph.

Find the point at $x = 2$ .

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

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

Simplify the result.

Simplify each term.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $π$ by $1$ .

$f (2) = 5 \sin (π - π) + 3$

Subtract $π$ from $π$ .

$f (2) = 5 \sin (0) + 3$

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

$f (2) = 5 \cdot 0 + 3$

Multiply $5$ by $0$ .

$f (2) = 0 + 3$

Add $0$ and $3$ .

$f (2) = 3$

The final answer is $3$ .

$3$

Find the point at $x = 3$ .

Replace the variable $x$ with $3$ in the expression.

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

Simplify the result.

Simplify each term.

Move $3$ to the left of $π$ .

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

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

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

Combine $- π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $- 1$ .

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

Subtract $2 π$ from $3 π$ .

$f (3) = 5 \sin (\frac{π}{2}) + 3$

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

$f (3) = 5 \cdot 1 + 3$

Multiply $5$ by $1$ .

$f (3) = 5 + 3$

Add $5$ and $3$ .

$f (3) = 8$

The final answer is $8$ .

$8$

Find the point at $x = 4$ .

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

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

Simplify the result.

Simplify each term.

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $π (4)$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $π \cdot (2)$ by $1$ .

$f (4) = 5 \sin (π \cdot (2) - π) + 3$

Move $2$ to the left of $π$ .

$f (4) = 5 \sin (2 π - π) + 3$

Subtract $π$ from $2 π$ .

$f (4) = 5 \sin (π) + 3$

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

$f (4) = 5 \sin (0) + 3$

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

$f (4) = 5 \cdot 0 + 3$

Multiply $5$ by $0$ .

$f (4) = 0 + 3$

Add $0$ and $3$ .

$f (4) = 3$

The final answer is $3$ .

$3$

Find the point at $x = 5$ .

Replace the variable $x$ with $5$ in the expression.

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

Simplify the result.

Simplify each term.

Move $5$ to the left of $π$ .

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

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

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

Combine $- π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $- 1$ .

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

Subtract $2 π$ from $5 π$ .

$f (5) = 5 \sin (\frac{3 π}{2}) + 3$

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.

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

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

$f (5) = 5 (- 1 \cdot 1) + 3$

Multiply $- 1$ by $1$ .

$f (5) = 5 \cdot - 1 + 3$

Multiply $5$ by $- 1$ .

$f (5) = - 5 + 3$

Add $- 5$ and $3$ .

$f (5) = - 2$

The final answer is $- 2$ .

$- 2$

Find the point at $x = 6$ .

Replace the variable $x$ with $6$ in the expression.

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

Simplify the result.

Simplify each term.

Cancel the common factor of $6$ and $2$ .

Factor $2$ out of $π (6)$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $π \cdot (3)$ by $1$ .

$f (6) = 5 \sin (π \cdot (3) - π) + 3$

Move $3$ to the left of $π$ .

$f (6) = 5 \sin (3 π - π) + 3$

Subtract $π$ from $3 π$ .

$f (6) = 5 \sin (2 π) + 3$

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

$f (6) = 5 \sin (0) + 3$

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

$f (6) = 5 \cdot 0 + 3$

Multiply $5$ by $0$ .

$f (6) = 0 + 3$

Add $0$ and $3$ .

$f (6) = 3$

The final answer is $3$ .

$3$

List the points in a table.

$\begin{array}{c} x & f (x) \\ 2 & 3 \\ 3 & 8 \\ 4 & 3 \\ 5 & - 2 \\ 6 & 3 \end{array}$

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

Amplitude: $5$

Period: $4$

Phase Shift: $2$ ( $2$ to the right)

Vertical Shift: $3$

$\begin{array}{c} x & f (x) \\ 2 & 3 \\ 3 & 8 \\ 4 & 3 \\ 5 & - 2 \\ 6 & 3 \end{array}$

Graph f(x)=5sin(pi/2x-pi)+3

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