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Parent Functions and their Graphs

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More Graphs and PreCalculus Lessons

Videos, solutions, worksheets, games and activities 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.

Graphs of parent functions

The following table shows the transformation rules for functions. Scroll down the page for examples and solutions on how to use the transformation rules.

Transformation rules for graph functions

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 = x2 (quadratic)
y = x3 (cubic
y = x5
y = |x| (absolute)
y = square root(x)
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) = square root(x)
f(x) = cube root(x)
f(x) = |x|
7 parent functions with equations, graphs, domain, range and asymptotes
y = x
y = x2
y = square root(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.
y = x,
y = x2,
y = x3,
y = sqrt(x),
y = 1/x,
y = |x|,
x2 + y2 = 9 (circle),
y = bx

Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.

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

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