1. Chapter 2 Sections 1- 3 Functions and Graphs

2. Definition of a Relation A Relation is a mapping, or pairing, of input values with output. A set of ordered pairs is a relation.

3. The values that make up the set of input values are the domain or also independent variables. The values that make up the set of output values are the range or also dependent variables

4. A relation is a function provided there is exactly one output for each input Input Output -3 1 3 4 3 1 -2 Domain Range Given: (-3, 3), (1, 1), (3, 1), (4, -2)

5. It is not a function if at least one input has more than one output. Input Output -3 1 4 3 1 -2 Domain Range Given: (-3, 3), (1, 1), (1, -2), (4, 4) 4

6. To determine if a graph is a function, we perform the vertical line test. -- Yes, it is a function. -- No, it is not a function.

7. Vertical Line Test for Functions A relation is a function if and only if no vertical line intersects the graph of the relation at more than one point.

8. Vertical Line Test: 1.Draw a vertical line through the graph. 2. See how many times the vertical line intersects the graph. 3. Only Once – Pass (function) More than Once – Fail (not function)

9. Is this graph a function? Yes, this is a function because it passes the vertical line test. Only crosses at one point.

10. Is this graph a function? No, this is not a function because it does not pass the vertical line test. Crosses at more than one point.

11. The functions in the last two examples are linear functions because it is of the form y = mx + b Linear Function where m and b are constants The graph of a linear function is a line. By naming a function “f ” you can write the function using function notation. f(x) = mx + b Function Notation

12. Function Notation The Symbolic Form A truly excellent notation. It is concise and useful.

13. Output Value Member of the Range Dependent Variable These are all equivalent names for the y. Input Value Member of the Domain Independent Variable These are all equivalent names for the x. Name of the function

14. Example of Function Notation The f notation

15. Decide whether the function is linear. Evaluate the function when x = -2 1.f(x) = -x 2 – 3x + 5 2. g(x) = 2x + 6 f(-2) = g(-2) =

16. Slope can be expressed different ways:

17. Slope is sometimes referred to as the “rate of change” between 2 points.

18. Types of Slope Positive Negative Zero Undefined or No Slope

19. What is the slope of a horizontal line? The line doesn’t rise! All horizontal lines have a slope of 0. f(x) = 3

20. What is the slope of a vertical line? The line doesn’t run! All vertical lines have an undefined slope. x = -2

21. Slope Parallel lines Their slopes will be EQUAL. Perpendicular lines Their slopes will be the negative reciprocal of each other.

22. Are the two lines parallel? L 1 : through (-2, 1) and (4, 5) and L 2 : through (3, 0) and (0, -2) This symbol means Parallel

23. Write parallel, perpendicular or neither for the pair of lines that passes through (5, -9) and (3, 7) and the line through (0, 2) and (8, 3). This symbol means Perpendicular

24. In the Mojave Desert in California, temperatures can drop quickly from day to night. Suppose the temperature drops from 100ºF at 2 P.M. to 68ºF at 5 A.M. Find the average rate of change and use it to determine the temperature at 10 P.M. Average rate of change = At 10 P.M. the temperature will be 84ºF

25. The formula for Slope-Intercept Form is: ‘b’ is the y-intercept. ‘m’ is the slope. Graph using the y-intercept and slope. f(x) = mx + b f(x) = 2x + 1

26. Sometimes we must solve the equation for y before we can graph it. The constant, b = 3 is the y-intercept. The coefficient, m = -2 is the slope. f(x) = -2x + 3

27. The standard form of a linear equation Ax + By = C where A and B are not both 0 To find the y intercept, let x = 0 and solve for y. Ax + By = C To find the x intercept. let y = 0 and solve for x. Ax + By = C

28. Graphing Equations with Intercepts 1.Write the equation in standard form. 2.Find the x-intercept by letting y = 0 and solving for x. Use this x-intercept to plot the point where the line crosses the x-axis. 3.Find the y-intercept by letting x = 0 and solving for y. Use the y-intercept to plot the point where the line crosses the y- axis. 4.Draw a line through the two points.

29. YOU TRY Graph: 3x - 2y = 6

30. The equation of a vertical line cannot be written in slope-intercept form because the slope of a vertical line is undefined Every linear equation, however, can be written in standard form -even the equation of a vertical line.

31. Horizontal and Vertical Lines Horizontal Lines: The graph of f(x) = c is a horizontal line through (0, c) Vertical Lines: The graph of x = c is a vertical line through (c, 0) f(x) = 5 x = -3

32. Example Graph: x = 2 Graph: f(x) = -3