Section 1'5 A Library of Functions

1
Section 1.5 A Library of Functions
2
What you should learn

• How to identify the graph linear and squaring
  functions
• How to identify and graph cubic, square root, and
  reciprocal functions
• How to identify and graph step and other
  piecewise-defined functions
• How to recognize graphs of common functions

3
Linear Functions

• A linear function can be written in the form
• f(x) ax b
• If a m and b b
• y mx b
• Find the linear function that passes through (-2,
  3) and (4, 0)

4
Linear Functions Given Two Points

• Calculate slope
• Find b
• y mx b
• Plug in slope and one point
• Use the found m and b to write
• f(x) ax b

5
Linear Functions To Remember

• Constant

Domain All Real Numbers
Range 4
6
Linear Functions To Remember

• Identity

Domain All Real Numbers
Range All Real Numbers
7
Graph of the Squaring Function
8
Characteristics of the Squaring Function

• The domain of the function is the set of all real
  numbers.
• The range of the function is all nonnegative real
  numbers.
• Why not just say all positive real numbers?
• The function is even.
• Meaning what?
• The graph has an intercept at (0, 0).
• Is it an x or a y intercept?

9
Characteristics of the Squaring Function

• The graph is decreasing on the interval (-8,
  0) and increasing on the interval (0, 8).
• The graph is symmetric with respect to the
  y-axis.
• Why not just say even?
• The graph has a relative minimum at (0,0).
• Meaning what?

10
Graph of the Cubic Function
11
Characteristics of the Cubic Function

• The domain of the function is the set of all real
  numbers.
• The range of the function is all the set of all
  real numbers.
• The function is odd.
• Meaning what?
• The graph has an intercept at (0, 0).
• Is it an x or a y intercept?

12
Characteristics of the Cubic Function

• The graph is increasing on the interval (-8,
  8).
• The graph is symmetric with respect to the
  origin.
• Why not just say odd?
• Does the graph have a relative minimum?

13
Graph of the Square Root Function
14
Characteristics of the Square Root Function

• The domain of the function is the set of all
  nonnegative real numbers.
• Why not just say all positive real numbers?
• The range of the function is all the set of
  nonnegative real numbers.
• The graph has an intercept at (0, 0).
• Is it an x or a y intercept?
• The graph is increasing on the interval (0,
  8).

15
Graph of the Reciprocal Function
16
Characteristics of the Reciprocal Function

• The domain of the function is
• (-8, 0) U (0, 8).
• Why not just say all real numbers?
• The range of the function is
• (-8, 0) U (0, 8).
• How about intercepts?
• The graph has no intercepts?

17
Characteristics of the Reciprocal Function

• The function is odd.
• The graph is symmetric with respect to the
  origin.
• The graph is decreasing on the intervals
• (-8, 0) and (0, 8).

18
Graph of the Greatest Integer Function
19
Characteristics of the Greatest Integer Function

• The domain of the function is all real numbers.
• The range of the function is all integers
• The graph has a y-intercept at (0, 0)
• What about x-intercepts?
• The graph has x-intercepts on the interval 0,1)
• The graphs is constant between each pair of
  consecutive integers.
• The graph jumps vertically one unit at each
  integer value.

20
Graphing a Piecewise-Defined Function
Is the point (1, 3) a part of this graph?
21
Homework 1.5

• Page 55
• Learn Graphs a g, omit h
• Page 56
• 1 - 27 odd, 43, 45, 53 - 61 odd Identify type of
  function only