Understanding the Period and Horizontal Shift

Oct 13, 2025 by windailys.com 33 views

If the Graph of the Function $f(x)=a \tan (b x+c)$ Has a Period of $\frac{1}{4}$ and a Horizontal Shift of $\frac{1}{2}$ to the Right, Determine the Values of $a$ , $b$ , and $c$

The graph of a function is a visual representation of the relationship between the input and output values of the function. In this article, we will focus on the graph of the function $f(x)=a \tan (b x+c)$ , which has a period of $\frac{1}{4}$ and a horizontal shift of $\frac{1}{2}$ to the right. We will determine the values of $a$ , $b$ , and $c$ using the given information.

Understanding the Period and Horizontal Shift

The period of a function is the distance between two consecutive points on the graph that have the same value. In other words, it is the length of the interval between two consecutive points on the graph that have the same value. The horizontal shift of a function is the distance between the graph of the function and the x-axis.

For the function $f(x)=a \tan (b x+c)$ , the period is given as $\frac{1}{4}$ , which means that the distance between two consecutive points on the graph that have the same value is $\frac{1}{4}$ . The horizontal shift is given as $\frac{1}{2}$ to the right, which means that the graph of the function is shifted $\frac{1}{2}$ unit to the right.

Determining the Value of $b$

The period of a tangent function is given by $\frac{\pi}{b}$ . Since the period of the function $f(x)=a \tan (b x+c)$ is $\frac{1}{4}$ , we can set up the equation:

\frac{\pi}{b} = \frac{1}{4}

Solving for $b$ , we get:

b = 4\pi

Determining the Value of $c$

The horizontal shift of a tangent function is given by $\frac{c}{b}$ . Since the horizontal shift of the function $f(x)=a \tan (b x+c)$ is $\frac{1}{2}$ to the right, we can set up the equation:

\frac{c}{b} = \frac{1}{2}

Substituting the value of $b$ that we found earlier, we get:

\frac{c}{4\pi} = \frac{1}{2}

Solving for $c$ , we get:

c = 2\pi

Determining the Value of $a$

The value of $a$ is not affected by the period or horizontal shift of the function. Therefore, we can choose any value for $a$ . For simplicity, let's choose $a=1$ .

In this article, we determined the values of $a$ , $b$ , and $c$ for the function $f(x)=a \tan (b x+c)$ , which has a period of $\frac{1}{4}$ and a horizontal shift of $\frac{1}{2}$ to the right. We found that $b=4\pi$ , $c=2\pi$ , and $a=1$ .

Possible Values for $a$ , $b$ , and $c$

$a$ : any real number
$b$ : $4\pi$
$c$ : $2\pi$

Example

Suppose we want to graph the function $f(x)=\tan (4\pi x+2\pi)$ . Using the values of $a$ , $b$ , and $c$ that we found earlier, we can graph the function as follows:

Graph the function $f(x)=\tan (4\pi x)$ .
Shift the graph of the function $f(x)=\tan (4\pi x)$ to the right by $\frac{1}{2}$ unit.

The resulting graph will be the graph of the function $f(x)=\tan (4\pi x+2\pi)$ .

Applications

The values of $a$ , $b$ , and $c$ that we found in this article have several applications in mathematics and science. For example, they can be used to model periodic phenomena, such as the motion of a pendulum or the vibration of a string.

References

[1] "Graphing Functions" by Math Open Reference
[2] "Tangent Function" by Wolfram MathWorld

Glossary

Period: the distance between two consecutive points on the graph that have the same value.
Horizontal Shift: the distance between the graph of the function and the x-axis.
Tangent Function: a trigonometric function that is defined as the ratio of the sine and cosine functions.
Q&A: If the Graph of the Function $f(x)=a \tan (b x+c)$ Has a Period of $\frac{1}{4}$ and a Horizontal Shift of $\frac{1}{2}$ to the Right, Determine the Values of $a$ , $b$ , and $c$

Q: What is the period of a tangent function?

A: The period of a tangent function is given by $\frac{\pi}{b}$ .

Q: How do you determine the value of $b$ for a tangent function with a period of $\frac{1}{4}$ ?

A: To determine the value of $b$ , you can set up the equation $\frac{\pi}{b} = \frac{1}{4}$ and solve for $b$ . This will give you the value of $b$ that corresponds to a period of $\frac{1}{4}$ .

Q: What is the horizontal shift of a tangent function?

A: The horizontal shift of a tangent function is given by $\frac{c}{b}$ .

Q: How do you determine the value of $c$ for a tangent function with a horizontal shift of $\frac{1}{2}$ to the right?

A: To determine the value of $c$ , you can set up the equation $\frac{c}{b} = \frac{1}{2}$ and solve for $c$ . This will give you the value of $c$ that corresponds to a horizontal shift of $\frac{1}{2}$ to the right.

Q: What is the value of $a$ for a tangent function?

A: The value of $a$ is not affected by the period or horizontal shift of the function. Therefore, you can choose any value for $a$ . For simplicity, let's choose $a=1$ .

Q: How do you graph a tangent function with a period of $\frac{1}{4}$ and a horizontal shift of $\frac{1}{2}$ to the right?

A: To graph a tangent function with a period of $\frac{1}{4}$ and a horizontal shift of $\frac{1}{2}$ to the right, you can follow these steps:

Graph the function $f(x)=\tan (4\pi x)$ .
Shift the graph of the function $f(x)=\tan (4\pi x)$ to the right by $\frac{1}{2}$ unit.

Q: What are some applications of the values of $a$ , $b$ , and $c$ that we found in this article?

A: The values of $a$ , $b$ , and $c$ that we found in this article have several applications in mathematics and science. For example, they can be used to model periodic phenomena, such as the motion of a pendulum or the vibration of a string.

Q: What are some common mistakes to avoid when working with tangent functions?

A: Some common mistakes to avoid when working with tangent functions include:

Not taking into account the period and horizontal shift of the function.
Not using the correct values of $a$ , $b$ , and $c$ .
Not graphing the function correctly.

Q: How can I practice working with tangent functions?

A: You can practice working with tangent functions by:

Graphing tangent functions with different periods and horizontal shifts.
Solving problems that involve tangent functions.
Using online resources, such as graphing calculators or math software, to explore tangent functions.

Q: What are some resources that I can use to learn more about tangent functions?

A: Some resources that you can use to learn more about tangent functions include:

Online tutorials and videos.
Math textbooks and workbooks.
Online communities and forums.

Q: How can I apply the concepts of tangent functions to real-world problems?

A: You can apply the concepts of tangent functions to real-world problems by:

Modeling periodic phenomena, such as the motion of a pendulum or the vibration of a string.
Analyzing data that involves tangent functions.
Using tangent functions to solve problems in fields such as physics, engineering, and economics.