Find Amplitude, Period, and Phase Shift y=1.5sin(8x)

Solution:

Step 1:

Identify the coefficients in the standard sine function $y=a\sin (bx-c)+d$ to determine amplitude, period, phase shift, and vertical shift.

• Amplitude coefficient $a=1.5$
• Period coefficient $b=8$
• Phase shift coefficient $c=0$
• Vertical shift $d=0$

Step 2:

Calculate the amplitude by taking the absolute value of $a$.

Amplitude: $\left|1.5\right|=1.5$

Step 3:

Determine the period of the sine function $y=1.5\sin (8x)$.

Step 3.1:

The formula for the period is $\frac{2\pi }{\left|b\right|}$.

Step 3.2:

Substitute the value of $b$ with $8$.

Period: $\frac{2\pi }{\left|8\right|}$

Step 3.3:

Compute the absolute value of $8$, which is $8$ itself.

Period: $\frac{2\pi }{8}$

Step 3.4:

Simplify the fraction by reducing common factors.

Step 3.4.1:

Extract the factor of $2$ from the numerator.

Period: $\frac{2(\pi )}{8}$

Step 3.4.2:

Eliminate the common factor.

Step 3.4.2.1:

Factor out $2$ from the denominator.

Period: $\frac{2\pi }{2\cdot 4}$

Step 3.4.2.2:

Cancel out the common factor of $2$.

Period: $\frac{\cancel{2}\pi }{\cancel{2}\cdot 4}$

Step 3.4.2.3:

Finalize the expression for the period.

Period: $\frac{\pi }{4}$

Step 4:

Compute the phase shift with the formula $\frac{c}{b}$.

Step 4.1:

The phase shift is found using $\frac{c}{b}$.

Phase Shift: $\frac{c}{b}$

Step 4.2:

Insert the values for $c$ and $b$.

Phase Shift: $\frac{0}{8}$

Step 4.3:

Calculate the result of dividing $0$ by $8$.

Phase Shift: $0$

Step 5:

Compile the characteristics of the sine function.

• Amplitude: $1.5$
• Period: $\frac{\pi }{4}$
• Phase Shift: $0$ (no shift)
• Vertical Shift: $0$ (no shift)

Step 6:

There is no step 6 provided in the original solution.

Knowledge Notes:

The sine function $y=a\sin (bx-c)+d$ is a trigonometric function where:

• $a$ represents the amplitude, which is the peak value of the function.

• $b$ affects the period of the function, which is the distance required for one complete cycle of the sine wave. The period is calculated as $T=\frac{2\pi }{\left|b\right|}$.

• $c$ determines the phase shift, which is the horizontal shift of the function. The phase shift is calculated as $\frac{c}{\left|b\right|}$.

• $d$ represents the vertical shift, moving the function up or down on the graph.

The absolute value, denoted by $\left|x\right|$, is the distance of a number $x$ from zero on the number line, without considering direction. It is always non-negative.

Simplifying fractions involves finding common factors in the numerator and denominator and canceling them out to reduce the fraction to its simplest form.

In the given function $y=1.5\sin (8x)$, there is no phase shift or vertical shift, as $c$ and $d$ are both zero. The amplitude is $1.5$, and the period is $\frac{\pi }{4}$, which are the key characteristics of this sine function.