1. 11.5 Graphs of Equations 11.6 Introduction to Functions 11.7 Function Notation

2. 11.5 Graphs of Equations We already know how to graph lines using several methods – Plotting Points Method – Intercept Method – Slope Intercept Method In this section we will investigate a relationship between different lines. We will then discuss a type of nonlinear graph

3. If we graph the line above it would look like the following. Notice that because there is no “b” term or “y-intercept” shown then the line passes through the origin.

5. This concept can be used in all types of graphs (linear or nonlinear) Right now we know mostly about the lines of the form y = mx + b If we let the original line be y = mx – Which is a line with some slope m, and it goes through (0,0) And then we change that graph by adding or subtracting a value of “b” we will then shift the y=mx line up or down. If b is positive we will shift it up “b” units, If b is negative we will shift it down “b” units.

6. Examples

7. In order to graph this equation we will use the plotting points method. XY -3 -2 0 1 2 3

9. Basic rules for sliding y=x 2 up and down

10. An investigation of ordered pairs can validate that only the y-values are impacted. We will use our calculators to expedite the process of finding ordered pairs.

11. Another graph: y = |x| The absolute value graph takes every input value and makes it positive. Check some points. XY -3 -2 0 1 2 3

12. What do you think the graph of y = |x|+3 will look like?

13. What about y = |x|-3

14. Turn to page 792 we will do 23 and 31 together, at least the x and y intercept parts needed for graphing.

15. 11.6 Introduction to Functions

16. Remember that a RELATION is a rule that pairs x-coordinates with y- coordinates. The following shows the prices of used Ford Mustangs that were listed in a newspaper. Looking at the year of 2009 there are three different prices associated with the year. This data is considered a relation.

17. Two definitions of Relations A relation is a rule that pairs each element in one set, called the domain, with one or more elements from a second set, called the range. A relation is a set of ordered pairs. The set of all first coordinates is the domain, the second set of coordinates is the range.

18. In a relation each x value can be paired with any y- value a particular x-value can be associated with one or many different y-values. A FUNCTION is a relation in which each x-value is paired with only one y-value. – So looking back at the Ford Mustang table/graph, because the x-coordinate of 2009 was paired with 3 different y-values we would say that this scenario is NOT a function

19. Determine a function from a list of ordered pairs. {(3,3), (1,-1), (0,-3), (4,6)} {(0,0), (1,1), (1,-1), (4,2), (4,-2)}

20. Function as a Mapping 1 -2 3 5 3 -5 7 12 This mapping represents a function. WHY? Because each ‘x value’ has exactly one corresponding ‘y value’

21. 1 -2 2 5 3 -3 1 9 This mapping is not a function. WHY NOT? There are some ‘x values’ that map or correspond to two different ‘y values’. So because one value in the domain results multiple values in the range then the mapping is not a function.

22. How can I think about a function? A function is an equation that takes an input value and gives an output value… It is like a machine that makes fruit juice… -the fruit put in is called the input -the juice that comes out is called the output The actual machine would be the function

23. Input  FRUIT MACHINEMACHINE Output  JUICE

24. Now lets think of that machine as an equation. Input  5 f(x) Output  14 f(x) = 2x + 4 f(x) = 2( 5 ) + 4 input values come from the domain, which are x values. Output values are the y values that we get.

25. Input  5 f(x) Output  14 f(x) = 2x + 4 f(x) = 2(5) + 4 input values come from the domain, which are x values. Output values are the y values that we get. NOTICE no matter what value we plug into our “machine”/function, we will always get a different number as an output. We will never plug 5 into this function and get something different than 14

26. Recall/Review A function takes in an ‘x value’ and produces a ‘y value’… Every ‘x value’ has a unique ‘y value’ For example… f(x) = 3x – 1 if I choose x to be 3, then my function produces 8. Plugging in 3 again will never give a number other than 8, EVER.

27. Looking at a graph We can look at a graph and determine whether it represents a function or not. We will use what is called the VERTICAL LINE TEST

28. Vertical Line Test If you can draw a vertical line through any part of a graph, and the line only intersects the graph one time, then the graph represents a function. If it intersects that graph two times or more, then the graph is not a representation of a function. Why does this work?

29. Vertical line test at work Y N Y N

30. Notice that if we construct a vertical line at an x value of 3, then that line intersects the graph at 2 different points. That means that when x is 3, y is 2.5 and y is also -2.5. But we know in order for something to be a function, if we plug x in, then we get only one y value back, here if we plug x in we get two y values back. That is why this is not a function.

31. 11.7 Function Notation

32. Function Notation y = 2x – 1 f(x) = 2x - 1 Normal equationFunction Notation This is read as “the function of x” or “f of x”

33. Evaluating Functions at X. If f(x) = 3x + 2, evaluate or find each value. a.) f(2) (wherever you saw an x in your f(x) replace it now with a 2) f(2) = 3(2) + 2 f(2) = 6 + 2 f(2) = 8 Once you have completed the evaluation the input number was 2 and the output number is 8. The input is the domain (x) and the output is the range (y). So now we have an ordered pair for this particular function.

34. If f(x) = 2x – 2, find/evaluate a.) f(-3) f(-3) = 2(-3) – 2 f(-3) = -6 -2 f(-3) = -8

35. If f(x) = x 2 + 2x – 3 evaluate/find a.) f(2) f(2) = (2) 2 + 2(2) – 3 f(2) = 4 + 4 – 3 f(2) = 5

36. Example similar to homework

37. If f={(2,8), (-4,0), (3,.5), (0,9)} and g={(2,2), (-4,4), (.5,0)} Evaluate the following f(2) g(.5) f(g(.5)) f(-4)-g(2)

38. Evaluate the following f(g(2)) g(f(2)) f(f(-2)) g(g(f(0)))