Find Amplitude, Period, and Phase Shift y=5sin(4x-2)-3
The problem provided requires analyzing a trigonometric function, specifically a sine function, to determine three of its characteristics. These characteristics are:
Amplitude - This measures the height of the wave, from the middle of the wave to the top or bottom. It is generally the coefficient in front of the sine function.
Period - This determines how long it takes for the wave to complete one full cycle. The period of a sine function is related to the coefficient of the variable within the sine function.
Phase Shift - This indicates the horizontal shift of the wave from the origin. It is affected by the constant term that is being subtracted or added to the variable within the sine function.
The question asks to calculate these three properties for the given sine function.
$y = 5 sin \left(\right. 4 x - 2 \left.\right) - 3$
Answer
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:
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.
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.
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.
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.
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.
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.
