Compare The Parent Functions In The Following Answer Choices. Which Parent Function Is Periodic And Has An Intercept At \[$(0,1)\$\]?A. \[$y = \cos X\$\]B. \[$y = 10^x\$\]C. \[$y = \sin X\$\]D. \[$y = \tan X\$\]

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Introduction

In mathematics, parent functions are the basic functions from which other functions are derived. They are used to create various transformations and are essential in understanding the behavior of different functions. In this article, we will compare the parent functions given in the answer choices and determine which one is periodic and has an intercept at (0,1).

Parent Functions

A. y = cos x

The parent function y = cos x is a trigonometric function that represents the cosine of an angle. It is a periodic function with a period of 2π, meaning it repeats itself every 2π units. The graph of y = cos x has a maximum value of 1 and a minimum value of -1.

B. y = 10^x

The parent function y = 10^x is an exponential function that represents the power of 10 raised to the power of x. It is not a periodic function, meaning it does not repeat itself at regular intervals. The graph of y = 10^x is a continuous and increasing function that passes through the point (0,1).

C. y = sin x

The parent function y = sin x is a trigonometric function that represents the sine of an angle. It is a periodic function with a period of 2π, meaning it repeats itself every 2π units. The graph of y = sin x has a maximum value of 1 and a minimum value of -1.

D. y = tan x

The parent function y = tan x is a trigonometric function that represents the tangent of an angle. It is a periodic function with a period of π, meaning it repeats itself every π units. The graph of y = tan x has a vertical asymptote at x = π/2 and x = -π/2.

Comparing Periodic Properties

From the above descriptions, we can see that options A and C are periodic functions, while option D is also periodic but with a different period. Option B is not a periodic function.

Comparing Intercept Properties

To determine which function has an intercept at (0,1), we need to evaluate each function at x = 0.

  • Option A: y = cos x has an intercept at (0,1) since cos(0) = 1.
  • Option B: y = 10^x has an intercept at (0,1) since 10^0 = 1.
  • Option C: y = sin x has an intercept at (0,0) since sin(0) = 0.
  • Option D: y = tan x is undefined at x = 0 since tan(0) is undefined.

Conclusion

Based on the comparison of the parent functions, we can conclude that:

  • Option A, y = cos x, is a periodic function with a period of 2π and has an intercept at (0,1).
  • Option B, y = 10^x, is not a periodic function but has an intercept at (0,1).
  • Option C, y = sin x, is a periodic function with a period of 2π but does not have an intercept at (0,1).
  • Option D, y = tan x, is a periodic function with a period of π but is undefined at x = 0.

Q: What are parent functions?

A: Parent functions are the basic functions from which other functions are derived. They are used to create various transformations and are essential in understanding the behavior of different functions.

Q: What are some examples of parent functions?

A: Some examples of parent functions include:

  • Trigonometric functions: sin x, cos x, tan x
  • Exponential functions: y = 10^x, y = e^x
  • Polynomial functions: y = x^2, y = x^3
  • Rational functions: y = 1/x, y = x/2

Q: What is the difference between a parent function and a transformed function?

A: A parent function is the basic function from which other functions are derived, while a transformed function is a function that has been modified in some way, such as a shift, stretch, or reflection.

Q: How do I determine if a function is periodic?

A: To determine if a function is periodic, look for the following characteristics:

  • The function repeats itself at regular intervals.
  • The function has a period, which is the length of time it takes for the function to repeat itself.
  • The function has a graph that is symmetrical about a vertical line.

Q: What is the period of a function?

A: The period of a function is the length of time it takes for the function to repeat itself. For example, the period of the sine function is 2π, meaning that the sine function repeats itself every 2π units.

Q: How do I find the intercepts of a function?

A: To find the intercepts of a function, evaluate the function at x = 0 and y = 0. The intercepts are the points where the function crosses the x-axis and y-axis.

Q: What is the difference between a vertical asymptote and a horizontal asymptote?

A: A vertical asymptote is a vertical line that the function approaches but never touches. A horizontal asymptote is a horizontal line that the function approaches as x approaches infinity or negative infinity.

Q: How do I determine if a function has a vertical or horizontal asymptote?

A: To determine if a function has a vertical or horizontal asymptote, look for the following characteristics:

  • Vertical asymptote: The function approaches a vertical line as x approaches a certain value.
  • Horizontal asymptote: The function approaches a horizontal line as x approaches infinity or negative infinity.

Q: What is the significance of parent functions in mathematics?

A: Parent functions are significant in mathematics because they are used to create various transformations and are essential in understanding the behavior of different functions. They are also used to model real-world phenomena and are a fundamental concept in calculus and other areas of mathematics.

Q: How do I use parent functions to model real-world phenomena?

A: To use parent functions to model real-world phenomena, identify the type of function that best represents the phenomenon and then apply transformations to the function to fit the data. For example, you can use the sine function to model the motion of a pendulum or the exponential function to model population growth.

Q: What are some common applications of parent functions?

A: Some common applications of parent functions include:

  • Modeling population growth and decay
  • Modeling the motion of objects
  • Modeling financial data
  • Modeling scientific data

Q: How do I choose the correct parent function to model a real-world phenomenon?

A: To choose the correct parent function to model a real-world phenomenon, identify the type of function that best represents the phenomenon and then apply transformations to the function to fit the data. Consider the following factors:

  • The type of data: Is it linear, quadratic, exponential, or trigonometric?
  • The behavior of the data: Is it increasing, decreasing, or oscillating?
  • The shape of the data: Is it smooth, jagged, or irregular?