Download College Algebra Chapter 2 Functions and Graphs

College Algebra
Chapter 2
Functions and Graphs
Section 2.7
Analyzing Graphs of
Functions and PiecewiseDefined Functions
Concepts
1. Test for Symmetry
2. Identify Even and Odd Functions
3. Graph Piecewise-Defined Functions
4. Investigate Increasing, Decreasing, and Constant
Behavior of a Function
5. Determine Relative Minima and Maxima of a
Function
Tests for Symmetry
Consider an equation in the variables x and y.
Symmetric with respect to the y-axis:
Substituting –x for x results in equivalent equation.
Symmetric with respect to the x-axis:
Substituting –y for y results in equivalent equation.
Symmetric with respect to the origin:
Substituting –x for x and –y for y results in equivalent
equation.
Example 1:
Determine whether the graph of the equation is
symmetric to the x-axis, y-axis, origin, or none of these.
y  x3
Example 2:
Determine whether the graph of the equation is
symmetric to the x-axis, y-axis, origin, or none of these.
y  x 5
2
Example 3:
Determine whether the graph of the equation is
symmetric to the x-axis, y-axis, origin, or none of these.
x 2  ( y  1)2  4
Example 4:
Determine whether the graph of the equation is
symmetric to the x-axis, y-axis, origin, or none of these.
y  x 2
Concepts
1. Test for Symmetry
2. Identify Even and Odd Functions
3. Graph Piecewise-Defined Functions
4. Investigate Increasing, Decreasing, and Constant
Behavior of a Function
5. Determine Relative Minima and Maxima of a
Function
Even and Odd Functions
Even function:
f(–x) = f(x) for all x in the domain of f.
(Symmetric to the y-axis)
Odd function:
f(–x) = –f(x) for all x in the domain of f.
(Symmetric to the origin)
Example 5:
Determine if the function is even, odd, or neither.
f ( x)  x 4  x 2  2
Example 6:
Determine if the function is even, odd, or neither.
f ( x)  x 3  x  3
Example 7:
Determine if the function is even, odd, or neither.
f ( x)  x 3  x
Example 8:
Determine if the function is even, odd, or neither.
Example 9:
Determine if the function is even, odd, or neither.
Concepts
1. Test for Symmetry
2. Identify Even and Odd Functions
3. Graph Piecewise-Defined Functions
4. Investigate Increasing, Decreasing, and Constant
Behavior of a Function
5. Determine Relative Minima and Maxima of a
Function
Example 10:
Evaluate the function for the given values of x.
 x2  1 x  0
f ( x)  
 x 1 x  0
f  1  _____
f  0   _____
f  5   _____
Example 11:
Evaluate the function for the given values of x.
 x2  3 x  2

h ( x )  7
2 x4
2 x  1 x  4

h 1  _____
h  3  ____
h  4   _____
h  5   _____
Example 12:
Graph the function.
1
 x 1
f ( x)   2

 x  1
x2
x2
Example 13:
Graph the function.
 1

z ( x)   x  1
2 x  6

x0
0 x2
x2
Greatest Integer Function
f ( x)  x
x
is the greatest integer less than or equal to x.
Example 14:
Evaluate.
1.3  ____
0.2  ____
5.6  ____
3  ____
Example 15:
Graph.
p ( x)  x  1
x
p ( x)  x  1
Example 16:
A new job offer in sales promises a base salary of $3000
a month. Once the sales person reaches $50,000 in total
sales, he earns his base salary plus a 4.3% commission
on all sales of $50,000 or more. Write a piecewisedefined function (in dollars) to model the expected total
monthly salary as a function of the amount of sales, x.
Concepts
1. Test for Symmetry
2. Identify Even and Odd Functions
3. Graph Piecewise-Defined Functions
4. Investigate Increasing, Decreasing, and Constant
Behavior of a Function
5. Determine Relative Minima and Maxima of a
Function
Intervals of Increasing, Decreasing, and Constant
Behavior
Increasing
Decreasing
Constant
Example 17:
Use interval notation to write the interval(s) over which
f ( x) is increasing, decreasing, and constant.
Increasing:
_____________________
Decreasing:
_____________________
Constant:
_____________________
Example 18:
Use interval notation to write the interval(s) over which
f ( x) is increasing, decreasing, and constant.
Increasing:
_____________________
Decreasing:
_____________________
Constant:
_____________________
Concepts
1. Test for Symmetry
2. Identify Even and Odd Functions
3. Graph Piecewise-Defined Functions
4. Investigate Increasing, Decreasing, and Constant
Behavior of a Function
5. Determine Relative Minima and Maxima of a
Function
Relative Minimum and Relative Maximum Values
Example 19:
Identify the location and value of any relative maxima
or minima of the function.
The point ________ is the
lowest point in a small interval
surrounding x = ____.
At x = ____ the function has a
relative minimum of _____.
Example 19 continued:
The point ________ is the
highest point in a small interval
surrounding x = ____.
At x = ____ the function has a
relative maximum of _____.
Example 20:
Identify the location and value of any relative maxima
or minima of the function.
At x = ____ the function has a
relative minimum of _____.
At x = ____ the function has a
relative minimum of _____.
At x = ____ the function has a
relative maximum of _____.
Example 21:
Identify the location and value of any relative maxima
or minima of the function.