Find Amplitude, Period, and Phase Shift y=1.5sin(8x)
The problem is asking for an analysis of a trigonometric function, specifically a sine function, to determine three characteristics:
Amplitude - This is the coefficient in front of the sine function, which determines the maximum height of the wave above the middle line, or the maximum depth below it. It represents half the distance between the maximum and minimum values of the function.
Period - The period refers to the length of one complete cycle of the sine wave. It is inversely related to the coefficient of the variable inside the sine function. For a sine function of the form sin(bx), the period is calculated by dividing the standard period of the sine function (2π for sine) by the absolute value of b.
Phase Shift - This is the horizontal shift of the sine wave along the x-axis. In a function of the form sin(bx-c) or sin(bx+c), the phase shift is determined by the value of c. It indicates how much the entire wave is moved to the right (for a positive c) or to the left (for a negative c) from the standard position of the sine function starting at the origin (0,0).
$y = 1.5 sin \left(\right. 8 x \left.\right)$
Answer
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.
