1. Lesson 3 & 4- Graphs of Functions
Unit 6
Lesson 3 & 4- Graphs of Functions

2. Objectives and HW Be able to graph a function by plotting points
Be able to graph piecewise defined functions
Be able to use the vertical line test
Vocabulary: linear function; constant function; greatest integer function/step function
HW: Read p
Do: p. 221: 2, 3, 4, 5, 11, 17, 23, 33, 35, 41, 49, 51, 53, 55, 57, 59

3. Review
What is the slope and y-intercept for the graph of the following equations:
y = -4x + 7
Ans: slope or m = -4; y-int = 7; (0,7)
y = 9 Hint: What does the graph look like?
Ans: This is a horizontal line; the slope = 0 and y-int = 9

4. New: Graphing Functions
There are a number of different types of functions you should be able to identify by either their equations or their graphs:
Linear functions: f(x) = mx + b
Constant function: f(x) = b
Power functions: f(x) = x2 or g(x) = x3
Root functions: f(x) = √x or g(x) =
Absolute value function: f(x) = |x|
Reciprocal functions: f(x) = 1/x or g(x) = 1/x2

5. New: Graphing Functions
The graphs for all the functions on the previous slide and more are on p. 221 of the text. You should familiarize yourself with the shapes and characteristics for each type of graph on p Not all of them will be reviewed in this lesson

6. New: Graphing Functions
To graph a linear function, simply find two points and draw the line. Ex. f(x) = 2x + 3 One point is the y-int: (0,3). A second point can be found by plugging in a value for x and finding f(x) f(1) = 5 so point (1,5) is on the graph or by using the slope  from point (0,3) go up 2 and to the right 1 to find another point on the line  (1,5)

7. New: Graphing Functions
A constant function is simply a horizontal line. All horizontal lines have a slope = 0 so their equations are f(x) = b in which b is the y-int Ex. f(x) = 7. To graph, it is the horizontal line through point (0,7)

8. New: Graphing Functions
For the other functions, it is helpful to know something about the shapes of their graphs as well as their domains and ranges. Even when plugging in values for x and finding f(x), it is helpful to know the general shape and how to determine the domain and range

9. New: Power Functions
For even power functions such as f(x) = x2 or f(x) = x4 : Domain: any value can be plugged in for x since there is no danger of having a denominator of 0 or a negative radical; domain is all real numbers Range: f(x) can never be negative since x is being raised to an even power (recall a negative times a negative is positive). The range has to be x ≥ 0 (cont to next slide)

10. New: Power Functions
Graph: You can use to see the graph. X2 is expressed as x^2 In general, you should know that the simplest even power function, f(x) = x2, has a parabola for its graph. All even power graphs have a similar shape

11. New: Power Functions
For odd power functions, such as f(x) = x3 or f(x) = x5: Domain: As with even power functions, there is no concern about denominators of 0 or negative roots. The domain is all real numbers Range: Unlike even powers, f(x) can be negative. For example, (-2)3 = -8. So, the range is all real #’s

12. New: Power Functions
Graph: Again, you can use fooplot.com to play with the graphs of odd power functions. You should be familiar with the simplest odd power function, f(x) = x3. When x is positive, f(x) is also positive, and when x is negative, f(x) is negative. How does it compare to the graphs for even power functions?

13. New: Root Functions
For even root functions, such as √x or : Domain: any negative even root has no real solutions; for example, √-4 is 2i which is an imaginary number. So, the domain has to be x ≥ 0 Range: f(x) has to be a positive since the domain is all positive; the range is all real #’s

14. New: Root Functions
Graph: On fooplot, remember that you can express radicals as rational exponents. For example, √x is the same as x^(1/2). For even root functions, the domain and range are both positive so the graph is entirely in Quadrant I

15. New: Root Functions
For odd root functions, such as : Domain: any real number works because it is an odd root. For example, = -2. The radicand can be negative in this case. Range: As with odd power functions, with odd root functions f(x) can be any real number

16. New: Root Functions
Graph: Use fooplot and/or p. 221 to help answer the questions below: 1. How does the graph for odd root functions compare to the graph for even root functions? 2. How does it compare to the graph for odd power functions? Check your answers with Mr. MacMillan or another student who has completed this lesson

17. New: Graphing Piecewise Functions
Graphs of piecewise functions can be thought of as parts of two or more graphs plotted on the same coordinate plane. The part of the graph to be plotted is dictated by the stated domain for each “piece” See Mr. MacMillan or another student who has completed this lesson for examples of piecewise graphs

18. New: Vertical Line Test
Definition- The vertical line test simply states that if a vertical line can be drawn through a graph and intersect it in two or more points, then it is not a graph of a function. Recall that a function means every “x” value is paired with one and only one “y” value. Any two points on a vertical line have the same x-coordinate but different y-coordinates so one x value would be paired with more than one y value Check to see if all the graphs on p. 221 pass the vertical line test