Find Amplitude, Period, and Phase Shift y=5sec(2x)
The problem involves analyzing a trigonometric function, specifically the secant function in the form y = A*sec(Bx), where 'A' and 'B' are constants that affect the amplitude, period, and phase shift of the function.
Amplitude: In the context of the secant function, 'amplitude' typically refers to the vertical stretch or compression factor. For the secant function, since it has no maximum or minimum value and instead has vertical asymptotes, the concept of amplitude does not apply in the traditional sense as it would for sine or cosine functions. However, the coefficient 'A' in front of the secant does indicate how it is vertically stretched or compressed.
Period: The period of a trigonometric function is the length of one complete cycle of the function. For secant functions, this is related to how quickly or slowly the function repeats its pattern. The period of a basic secant function is π, and this is modified by the coefficient 'B' as the formula for the period becomes the absolute value of (π/B).
Phase shift: The phase shift refers to the horizontal displacement of the function. It indicates how the basic function y = sec(x) is shifted left or right on the x-axis. In the given function y = 5*sec(2x), there is no explicit horizontal shift mentioned, so you might infer that there is no phase shift. If there were an addition or subtraction inside the argument of the secant, that would represent the phase shift.
The question asks you to compute the period and the phase shift of the function based on the given coefficients, and to understand the effect of coefficient 'A' on the function's amplitude-like vertical scaling.
$y = 5 sec \left(\right. 2 x \left.\right)$
Answer
Solution:
Step 1:
Identify the coefficients and constants in the equation $y = a \sec(bx - c) + d$ to determine amplitude, period, phase shift, and vertical shift.
- Amplitude coefficient ($a$): $5$
- Period coefficient ($b$): $2$
- Phase shift constant ($c$): $0$
- Vertical shift constant ($d$): $0$
Step 2:
The secant function, $\sec(x)$, does not have a defined amplitude as it extends to infinity.
- Amplitude: Not applicable
Step 3:
Calculate the period of the secant function $y = 5\sec(2x)$.
Step 3.1:
Use the formula for period $P = \frac{2\pi}{|b|}$.
Step 3.2:
Substitute $b = 2$ into the period formula: $P = \frac{2\pi}{|2|}$.
Step 3.3:
Compute the absolute value of $2$, which is $2$: $P = \frac{2\pi}{2}$.
Step 3.4:
Simplify the expression by reducing the fraction.
Step 3.4.1:
Eliminate the common factor of $2$: $P = \frac{\cancel{2}\pi}{\cancel{2}}$.
Step 3.4.2:
The period is $\pi$: $P = \pi$.
Step 4:
Determine the phase shift using the formula $\frac{c}{b}$.
Step 4.1:
The phase shift is computed as: Phase Shift = $\frac{c}{b}$.
Step 4.2:
Insert the values for $c$ and $b$: Phase Shift = $\frac{0}{2}$.
Step 4.3:
Calculate the phase shift: Phase Shift = $0$.
Step 5:
Summarize the properties of the function.
- Amplitude: Not applicable
- Period: $\pi$
- Phase Shift: $0$ (no shift)
- Vertical Shift: $0$ (no shift)
Knowledge Notes:
The secant function, $\sec(x)$, is the reciprocal of the cosine function, $\cos(x)$, and is defined as $\sec(x) = \frac{1}{\cos(x)}$. The secant function has the following properties:
Amplitude: The secant function does not have a maximum or minimum value, as it approaches infinity. Therefore, it does not have an amplitude.
Period: The period of the secant function is the length of one complete cycle on the graph. For the function $y = a \sec(bx - c) + d$, the period is calculated using the formula $P = \frac{2\pi}{|b|}$, where $b$ is the coefficient of $x$.
Phase Shift: The phase shift of a trigonometric function is the horizontal shift along the x-axis. For the function $y = a \sec(bx - c) + d$, the phase shift is calculated using the formula $\frac{c}{b}$, where $c$ is the constant that is subtracted from $bx$.
Vertical Shift: The vertical shift is the amount by which the graph of the function is moved up or down along the y-axis. For the function $y = a \sec(bx - c) + d$, the vertical shift is the constant $d$.
Absolute Value: The absolute value of a number is its distance from zero on the number line, denoted by $|x|$. It is always non-negative.
In the given problem, the secant function $y = 5 \sec(2x)$ has a period coefficient $b = 2$, which affects the period of the function. There is no phase shift or vertical shift because $c = 0$ and $d = 0$.
