# Understanding Trigonometric Functions: Period and Phase Shift Explained

• Jess048
In summary, the function y= -4 tan (1/2x + 3pie/8) has a period of pie and a phase shift of -3pie/8. The period is found by using the general formula given in the text or notes for this type of function, which states that the period of y = tan kx is pie/k where k is greater than 0. The phase shift is found by considering one period of the function and determining the starting value of x and the difference between the two values of x.

#### Jess048

State the period and phase shift of the function y= -4 tan (1/2x + 3pie/8).

I know the answer is pie; -3pie/8, but I don't understand the process could someone explain how the period and phase shift are found in these functions.

Could you share with us the general forumla given in your text or notes for this type of function?

Do you know the definitions of "phase shift" and period? What is the period of tan(x)?

The period of function y = tan k0 is pie/k where k>o. I'm not sure about the phase shift.

Maybe this'll help:
Consider y=x^2

y=(x-2.5)^2 is the same function, shifted to the right 2.5 units.

y=(2x-6)^2 would have to first be written as
y=(2(x-3))^2
This is the function y=x^2 shifted 3 units to the right. The 2 does something else to the function (stretches it vertically in this case.)
Can you get (x-#) in your problem?

tan(x) has period $\pi$. In particular, $tan(0)= tan(\pi)$.
One period starts at x= 0 and ends at $x= \pi$.

Okay, one period of $-4tan((1/2)x- 3\pi/8)$ "starts" when $(1/2)x- 3\pi/8= 0$ and ends when $(1/2)x- 3\pi/8= \pi$. What is the "starting" value of x (the phase shift) and what is the difference between the two values of x (the period)?

## 1. What is a trigonometric function?

A trigonometric function is a mathematical function that relates the angles of a triangle to the lengths of its sides. The most commonly used trigonometric functions are sine, cosine, and tangent.

## 2. What is the period of a trigonometric function?

The period of a trigonometric function is the length of one complete cycle of the function. For example, the period of the sine function is 2π, which means that the graph of the function repeats itself every 2π units.

## 3. How do you determine the period of a trigonometric function?

To determine the period of a trigonometric function, you can use the formula 2π/b, where b is the coefficient of the angle in the function. For example, the sine function f(x) = sin(2x) has a period of 2π/2 = π.

## 4. What is the phase shift of a trigonometric function?

The phase shift of a trigonometric function is the horizontal translation of the graph of the function. It represents how much the graph is shifted to the left or right compared to the standard position of the function. It is denoted by the letter c in the function f(x) = sin(ax + c).

## 5. How do you find the phase shift of a trigonometric function?

To find the phase shift of a trigonometric function, you can use the formula -c/a, where c is the coefficient of the angle and a is the coefficient of the variable in the function. For example, the cosine function f(x) = cos(3x - π/4) has a phase shift of -(π/4)/3 = -π/12.