# Parent Functions And Their Graphs

Lessons with videos, examples and solutions to help PreCalculus students learn how about parent functions and their graphs.

The following figures show the graphs of parent functions: linear, quadratic, cubic, absolute, reciprocal, exponential, logarithmic, square root, sine, cosine, tangent. Scroll down the page for more examples and solutions. The following table shows the transformation rules for functions. Scroll down the page for examples and solutions on how to use the transformation rules. #### Parent Functions And Their Graphs - How To Graph Elementary Functions?

In math, we often encounter certain elementary functions. These elementary functions include rational functions, exponential functions, basic polynomials, absolute values and the square root function. It is important to recognize the graphs of elementary functions, and to be able to graph them ourselves. This will be especially useful when doing transformations.

#### Basic Graphs That Every Algebra Student Should Know

Basic graphs that are useful to know for any math student taking algebra or higher.
y = mx + b (linear function)
y = x3 (cubic)
y = x5
y = |x| (absolute)
y = √x (square root)
y = 1/x (reciprocal)
y = 1/x2
y = logb(x) for b > 1
y = ax for a > 1 (exponential)
y = ax for 0 < a < 1

#### The Graphs Of Six Basic Functions That You Should Know

f(x) = x
f(x) = x2
f(x) = x3
f(x) = √x
f(x) = cube root(x)
f(x) = |x|

y = x
y = x2
y = √x
y = x3
y = 1/x
y = 1/x2
y = |x|

#### Exploring Properties Of Parent Functions

In math, every function can be classified as a member of a family. Each member of a family of functions is related to its simpler, or most basic, function sharing the same characteristics. This function is called the parent function.

This lesson discusses some of the basic characteristics of linear, quadratic, square root, absolute value and reciprocal functions.

#### Transformations Of Parent Functions

Learn how to shift graphs up, down, left, and right by looking at their equations.

Vertical Shifts:
f(x) + c moves up,
f(x) - c moves down.

Horizontal Shifts:
f(x + c) moves left,
f(x - c) moves right.

#### Transforming Graphs And Equations Of Parent Functions

Looking at some parent functions and using the idea of translating functions to draw graphs and write equations.

Here is a list of topics:

1. F(x) functions and transformations
2. Horizontal Shift - Left and Right Units
3. Vertical Shift - Units Up and Down
4. Reflection about the x-axis, y-axis, and origin
5. Inverse function f-1(x)
6. Domain and Range - X and Y Values
7. Horizontal and Vertical Asymptotes
8. End Behavior - Left and Right Side
9. Linear Functions: y=x
10. Parabolas and Quadratic Functions: y=x2
11. Polynomial Functions - Cubic Functions: y=x3
12. Absolute Value Functions
13. Cube Root Functions
14. Square Root and Radical Functions
15. Rational Functions y = 1/x - Vertical and Horizontal Asymptotes
16. Exponential Functions: y=ex
17. Logarithmic Functions - Log and Natural Log Functions y=lnx
18. Trigonometric Functions - sine, cosine, and tangent - sin cos tan
19. Inverse Trigonometric Functions y = sin-1 x cos-1(x) and tan-1(x)

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. 