Find Amplitude, Period, and Phase Shift y=5sec(2x)

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:

1. Amplitude: The secant function does not have a maximum or minimum value, as it approaches infinity. Therefore, it does not have an amplitude.

2. 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 $x$.

3. 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 $bx$.

4. 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 $d$.

5. 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 $d=0$.