Find Amplitude, Period, and Phase Shift y=5sin(4x-2)-3

Solution:

Step 1: Identify the coefficients from the standard sine function form $a\sin (bx-c)+d$

• Amplitude coefficient ($a$): $5$
• Frequency coefficient ($b$): $4$
• Phase shift coefficient ($c$): $2$
• Vertical shift ($d$): $-3$

Step 2: Determine the amplitude

• Amplitude is the absolute value of $a$: $\left|5\right|=5$

Step 3: Calculate the period

• The formula for the period of a sine function is $\frac{2\pi }{\left|b\right|}$

Step 3.1: Apply the formula to our function

• Substitute $b=4$ into the formula: $\frac{2\pi }{\left|4\right|}$

Step 3.1.1: Simplify the expression

• The absolute value of $4$ is $4$, so the expression becomes $\frac{2\pi }{4}$

Step 3.1.2: Reduce the fraction

• Divide both the numerator and the denominator by $2$: $\frac{\pi }{2}$

Step 4: Compute the phase shift

• Phase shift is found using the formula $\frac{c}{b}$

Step 4.1: Insert values into the phase shift formula

• With $c=2$ and $b=4$, the phase shift is $\frac{2}{4}$

Step 4.2: Simplify the phase shift

• Reduce the fraction by dividing both numerator and denominator by $2$: $\frac{1}{2}$

Step 5: Compile the function properties

• Amplitude: $5$
• Period: $\frac{\pi }{2}$
• Phase Shift: $\frac{1}{2}$ (to the right)
• Vertical Shift: $-3$

Knowledge Notes:

The problem involves analyzing a trigonometric function to determine its amplitude, period, phase shift, and vertical shift. Here are the relevant knowledge points:

1. Amplitude: The amplitude of a trigonometric function like $y=a\sin (bx-c)+d$ or $y=a\cos (bx-c)+d$ is the absolute value of the coefficient $a$. It represents the maximum value the function reaches above or below its midline.

2. Period: The period of a trigonometric function is the length of one complete cycle of the curve. For sine and cosine functions, the period is calculated using the formula $\frac{2\pi }{\left|b\right|}$, where $b$ is the coefficient of $x$ in the function. The period tells us how quickly the sine or cosine wave repeats itself.

3. Phase Shift: The phase shift of a function is the horizontal shift along the x-axis. For the sine and cosine functions, it is calculated using the formula $\frac{c}{b}$, where $c$ is the constant subtracted from $bx$ in the function. A positive phase shift means the function is shifted to the right, while a negative phase shift means it is shifted to the left.

4. Vertical Shift: The vertical shift is the value $d$ in the function $y=a\sin (bx-c)+d$ or $y=a\cos (bx-c)+d$. It moves the graph up or down on the y-axis.

5. Absolute Value: The absolute value of a number is its distance from zero on the number line, denoted as $\left|x\right|$. It is always a non-negative number.

6. Reducing Fractions: To simplify a fraction, divide the numerator and the denominator by their greatest common factor.

Understanding these concepts is crucial for analyzing and graphing trigonometric functions.