A Library of Parent Functions Digital Lesson

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Definition: Functions Linear Function : The graph of the linear function has the following Characteristics.  x – intercept : y – intercept :  The graph is increasing if, decreasing if, and constant if.  

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 Definition: Functions Example 1. Write the linear function f for which f(1)=3 and f(4)=0 2. Write the linear function f for which f(0)=-1 and f(5)=8 - Constant function - Identities function

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Definition: Functions Squaring Function : Is a U-shaped curve with the following characteristics.  The domain of the function is the set of all real numbers.  The range of the function is the set of all nonnegative real numbers.  The function is even.  The graph has an intercept at (0,0).  The graph is decreasing on the interval and increasing on the interval  The graph is symmetric with respect to the y-axis.  The graph has a relative minimum at (0,0)

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Definition: Functions Cubic Functions :  The domain of the function is the set of all real numbers.  The range of the function is the set of all real numbers.  The function is odd.  The graph has an intercept at (0,0).  The graph is increasing on the interval  The graph is symmetric with respect to the origin.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Definition: Functions Square Root Functions :  The domain of the function is the set of all nonnegative real numbers.  The range of the function is the set of all nonnegative real numbers.  The graph has an intercept at (0,0).  The graph is increasing on the interval

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Definition: Functions Reciprocal Functions :  The domain of the function is  The range of the function is  The function is odd.  The graph does not have any intercepts.  The graph is decreasing on the intervals and  The graph is symmetric with respect to the origin.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Definition: Functions Step Piecewise-Defined Functions Functions whose graphs resemble sets of stairsteps are known as step functions. The most famous of the step functions is the greatest integer function, which is denoted by [x] and defined as f (x) = [x] = the greatest integer less than or equal to x.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Definition: Functions  The domain of the function is the set of all real numbers.  The range of the function is the set of all integers.  The graph has a y-intercept at (0,0) and x-intercepts in the interval [0,1).  The graph is constant between each pair of consecutive integers.  The graph jumps vertically one unit at each integer value.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Definition: Functions Example 2. Sketch the graph of 1. Evaluate the function when

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Parent Functions