1. Graphs and Functions Chapter 5

2. Introduction  We will build on our knowledge of equations by relating them to graphs.  We will learn to interpret graphs and obtain meaning from them.  Learn and use functional notations.  Learn to model function rules with tables and graphs.  Identify direct variations and find constants of variations.

3. Relating Graphs to Events (5.1)  Equations, inequalities, and proportions are all used to make a statement about a variable.  Graphs can be used to show the relationship between two variables.

4. Relations and Functions (5.2)  Relation: A set of ordered pairs.  Domain: The set of first coordinates of the ordered pairs of a relation.  These are the “x-coordinates.”  The values should be listed in order and no values should be repeated.  Range: The set of second coordinates of the ordered pairs of a relation.  These are the “y-coordinates.”  The values should be listed in order and no values should be repeated.

5. Relations and Functions (5.2) Age (years)Height(meters) 184.25 204.40 215.25 145.00 184.85 Ages and Heights of 5 Giraffes The relation for this set of data is: (18, 4.25) (20, 4.40) (21, 5.25) (14, 5.00) (18, 4.85) The domain for this set of data is: {14, 18, 20, 21} The range for this set of data is: {4.25, 4.40, 4.85, 5.00, 5.25}

6. Relations and Functions (5.2)  Function: A relation that assigns exactly one value in the range to each value in the domain.  Each range value will has at least one unique domain value.  Use the vertical-line test to determine if a relation can serve as a function.  Vertical-line test:  Plot all the ordered pairs on a graph.  Analyze the graph to see if any two points can be connected by a vertical line.  If any two points are connected by this vertical line, then the relation is not a function.

7. Relations and Functions (5.2)  The vertical line test for a function.

8. Relations and Functions (5.2)  Use of a mapping diagram can also help to determine if a relation is a function.  Making a mapping diagram:  List domain and range values in separate columns in order and do not repeat values.  Draw arrows from the domain values to their range values.  If there is no value in the domain that corresponds to more than one value of the range the relation is a function.

9. Relations and Functions (5.2)  Mapping diagrams to determine a function. {(11, -2), (12, -1), (13, -2), (20, 7)} Domain 11 12 13 20 Range -2 7 No value in the domain that corresponds to more than one value of the range. Function

10. Relations and Functions (5.2)  Sample Problem Use a mapping diagram to determine whether each the following relation is a function. {(-2, -1), (-1, 0), (6,3), (-2, 1)}

11. Relations and Functions (5.2)  Function rule: An equation that describes a function.  Think of a function rule as an input-output machine.  Therefore, if we know the input values, we can use the function rule to determine the output values. Input The domain values. Input-Output Machine Output The range values.

12. Relations and Functions (5.2)  Function notation: A function is in function notation when we use the symbol f(x) to indicate the output values.  Example: f(x) = 3x +4  We read the notation f(x):  “f of x”  “f is a function of x”  Other function notations you could see are: g(x), h(x), etc.  The function notation is simply another way to write an equation.  F(x) corresponds to y.

13. Relations and Functions (5.2)  Sample Problem Evaluate f(x) = -3n – 10 for n = 6. Evaluate y =-2x 2 + 7 for x= -4

14. Relations and Functions (5.2)  Sample Problem Evaluate the function rule f(a) = -3a + 5 to find the range of the function for the domain {-3, 1, 4}

15. Functions Rules, Tables, and Graphs (5.3)  We can use rules, tables, and graphs to model functions.  Remember a function rule shows us how different variables are related.  A table identifies specific input and output values of the function.  A graph gives a visual picture of the function.  We will be using the coordinate plane graphs learned back in Chapter 1.9.

16. Functions Rules, Tables, and Graphs (5.3)  Graphing a function.  Independent variables: These are the input values or the given domain values.  Graph the independent values on the horizontal axis (the x-axis)  These independent variables will be the x-coordinates  Dependent Variable: These are the output values that are obtained or the range values.  Graph the dependent variables on the vertical axis (the y-axis)  These dependent variables will be the y-coordinates.  Use the input and output values as ordered pairs to plot points on the graph.  Trace line or a smooth curve through the data points to get a general picture of the function.

17. Functions Rules, Tables, and Graphs (5.3)

18.  Some functions have graphs that are not straight lines.  In these cases you can still graph the function as long as you have the function rule.  To graph these functions:  Evaluate the function rule for the given domain value to obtain the range values.  Plot the ordered pairs obtained.  Join the data points with a smooth line or curve.

19. Functions Rules, Tables, and Graphs (5.3)  Sample Problem  Graph the function y = │x│ +1 if you have the following domain {-3, -1, 0, 1, 3}.

20. Functions Rules, Tables, and Graphs (5.3)  Sample Problem  Graph the function f(x)= x 2 +1 if you have the following domain {-2, -1, 0, 1, 2}.

21. Writing a Function Rule (5.4)  We saw in the previous section that we can use a function rule to generate a graph of that function rule.  We can go in the reverse direction: that is we can obtain a function rule by analyzing a graph or table of that function rule.  This process starts by looking for any patterns that comes from relating the independent and dependent variables in tables and in graphs.  When writing the function, the dependent variable is defined in terms of the independent variable.  Using a graph is preferred over a table especially when many numbers are involved or the function is complex.

22. Writing a Function Rule (5.4)  Writing a function rule from data organized in a table requires that you try to find the relationship between the numbers in the data table. xf(x) 15 26 37 48 Ask: “What can I do to the first number get the second number?” Relate: f(x) equals x plus 4 Write: f(x) = x + 4

23. Direct Variation (5.5)

24.  Sample Problem Do the following equation represent direct variations? a) 5x + 2y = 0 b) 5x + 2y = 9

25. Direct Variation (5.5)  To write an equation for a direct variation:  Find the constant (k) for any other point beside (0,0)  Then use this value for k and the two points that were used to write the equation in the form of y = kx. Sample Problem: Write an equation of the direct variation that includes the point (4,-3).

26. Direct Variation (5.5) Deriving the above ratio.

27. Direct Variation (5.5) XY -32.25 1-0/75 4-3 6-4.5

28. Graphs and Functions Chapter 5 THE END