Parent Functions
What are parent functions?
The parent functions are a base of functions you should be able to recognize the graph of given the function and the other way around. For our course, you will be required to know the ins and outs of 15 parent functions.
The Parent FunctionsThe fifteen parent functions must be memorized. You must be able to recognize them by graph, by function, and be able to sketch them.
The parent functions are a base of functions you should be able to recognize the graph of given the function and the other way around. For our course, you will be required to know the ins and outs of 15 parent functions.
The Parent FunctionsThe fifteen parent functions must be memorized. You must be able to recognize them by graph, by function, and be able to sketch them.
Function Symmetry
Functions can have one of three symmetrical properties: even, odd or neither.
Read below to learn how to determine whether a function is even, odd or neither.
Even Symmetry
An even function has symmetry about the
y-axis. That is, you can take the half of the graph of the function, fold it in half on y-axis, and the two halves of the graph will line up perfectly. Example: f(x) = x² If the graph of this function were to be folded over the y-axis, the two halves of the parabola would line up one over the other. This means the graph has even symmetry.
Even Function Test A function y = f(x) is an even function if f(x) = f(-x). Applying the even function test, if we pick the x = 2, then -x implies -2. So for a function to be even, f(2) and f(-2) must have the same value. For this particular f, x², f(2) = 4 and f(-2) = 4. This means the function is even. |
Odd Symmetry
An odd function has symmetry about the origin. Being symmetric about the origin can be related to folding the graph of the function on the x- and y-axis and having the pieces of the graph match exactly.
Example: f(x) = x³ Odd symmetry can be determined in 2 ways looking at the graph of a function.
1. Take the portion of the graph in quadrant I and rotate it clockwise to quadrant III using the origin as the point of rotation. If the two pieces of the graph line up, the function is odd. 2. Fold the graph of the function over the x- and y-axes. If the two sections of the graph of the function lie on top of one another, the graph has odd symmetry. Odd Function Test A function y = f(x) is an even function if f(-x) = -f(x). Applying the even function test, if we pick the x = 2, then -x implies -2. So for a function to be even, f(2) and f(-2) must have the same value. For this particular f, x³, f(2) = 8 so -f(2) = -8. Then f(-2) = -8. So f(-2) = -f(2) and this means the function is odd. |
Neither Even nor Odd
(No Symmetry) A function is neither even nor odd if it does not have the characteristics of an even function nor an odd function.
Example: f(x) = ln(x) It is clear to see from the graph of the exponential function that it cannot be folded or rotated in any manner with respect to the x- or y-axes or the origin and produce symmetry.
Graphs of this kind have neither even nor odd symmetry - or no symmetry. |
Assignments
Quiz
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Practice: First, spend some time studying the parent functions and the information on function symmetry presented above. Work on memorizing the parent functions and identifying symmetries of graphs.
Document: Matching parent functions and identifying symmetry. Document: Sketching parent functions. Assignment: The assignment for this unit is below. Remember, this is not due by any date...ever. It's just extra practice for you to become more familiar with the material! Once you have completed and submitted the assignment above, you may take the quiz below. The quiz is not self grading, but will give you an idea of what types of questions I might ask about the unit circle in the future. You do not need to submit the quiz.
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