Find Amplitude, Period, and Phase Shift y=7sin(-(3pi)/4x-pi/4)+6

$y = 7 \sin (- \frac{3 π}{4} x - \frac{π}{4}) + 6$

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

$a = 7$

$b = - \frac{3 π}{4}$

$c = \frac{π}{4}$

$d = 6$

Find the amplitude $| a |$ .

Amplitude: $7$

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

Find the period of $7 \sin (- \frac{3 x π}{4} - \frac{π}{4})$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $- \frac{3 π}{4}$ in the formula for period.

$\frac{2 π}{| - \frac{3 π}{4} |}$

$- \frac{3 π}{4}$ is approximately $- 2.35619449$ which is negative so negate $- \frac{3 π}{4}$ and remove the absolute value

$\frac{2 π}{\frac{3 π}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{4}{3 π}$

Cancel the common factor of $π$ .

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{4}{3 π}$

Factor $π$ out of $3 π$ .

$π \cdot 2 \frac{4}{π \cdot 3}$

Cancel the common factor.

$π \cdot 2 \frac{4}{π \cdot 3}$

Rewrite the expression.

$2 (\frac{4}{3})$

Combine $2$ and $\frac{4}{3}$ .

$\frac{2 \cdot 4}{3}$

Multiply $2$ by $4$ .

$\frac{8}{3}$

Find the period of $6$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $- \frac{3 π}{4}$ in the formula for period.

$\frac{2 π}{| - \frac{3 π}{4} |}$

$- \frac{3 π}{4}$ is approximately $- 2.35619449$ which is negative so negate $- \frac{3 π}{4}$ and remove the absolute value

$\frac{2 π}{\frac{3 π}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{4}{3 π}$

Cancel the common factor of $π$ .

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{4}{3 π}$

Factor $π$ out of $3 π$ .

$π \cdot 2 \frac{4}{π \cdot 3}$

Cancel the common factor.

$π \cdot 2 \frac{4}{π \cdot 3}$

Rewrite the expression.

$2 (\frac{4}{3})$

Combine $2$ and $\frac{4}{3}$ .

$\frac{2 \cdot 4}{3}$

Multiply $2$ by $4$ .

$\frac{8}{3}$

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

$\frac{8}{3}$

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: $\frac{\frac{π}{4}}{- \frac{3 π}{4}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{π}{4} \cdot (- \frac{4}{3 π})$

Cancel the common factor of $π$ .

Move the leading negative in $- \frac{4}{3 π}$ into the numerator.

Phase Shift: $\frac{π}{4} \cdot \frac{- 4}{3 π}$

Factor $π$ out of $3 π$ .

Phase Shift: $\frac{π}{4} \cdot \frac{- 4}{π \cdot 3}$

Cancel the common factor.

Phase Shift: $\frac{π}{4} \cdot \frac{- 4}{π \cdot 3}$

Rewrite the expression.

Phase Shift: $\frac{1}{4} \cdot \frac{- 4}{3}$

Cancel the common factor of $4$ .

Factor $4$ out of $- 4$ .

Phase Shift: $\frac{1}{4} \cdot \frac{4 (- 1)}{3}$

Cancel the common factor.

Phase Shift: $\frac{1}{4} \cdot \frac{4 \cdot - 1}{3}$

Rewrite the expression.

Phase Shift: $\frac{- 1}{3}$

Move the negative in front of the fraction.

Phase Shift: $- \frac{1}{3}$

List the properties of the trigonometric function.

Amplitude: $7$

Period: $\frac{8}{3}$

Phase Shift: $- \frac{1}{3}$ ( $\frac{1}{3}$ to the left)

Vertical Shift: $6$

Find Amplitude, Period, and Phase Shift y=7sin(-(3pi)/4x-pi/4)+6

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