1. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 1 Chapter 1 Introduction to Functions and Graphs

2. 2 Copyright © 2014, 2010, 2006 Pearson Education, Inc. Visualizing and Graphing Data ♦ Analyze one-variable data ♦ Find the domain and range of a relation ♦ Graph in the xy-plane ♦ Calculate distance ♦ Find the midpoint ♦ Learn the standard equation of a circle ♦ Learn to graph equations with a calculator (optional) 1.2

3. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 3 Example: Analyzing a list of temperatures The table lists the low temperatures T in degrees Fahrenheit that occurred in Minneapolis, Minnesota, for six consecutive nights during January 2012. Temperature ºF–12–4–821189 (a)Plot these temperatures on a number line. (b)Find the maximum and minimum temperatures. (c)Determine the mean of these six temperatures. (d)Find the median and interpret the results.

4. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 4 Example: Example: Analyzing a list of temperatures (a) (b)Maximum (farthest to the right) is 21ºF Minimum (farthest to the left) is –12ºF (c) Mean: (d)Even number of data points, so the median is average of the middle two: –4 and 9; the median is –2.5ºF.

5. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 5 Two-Variable Data: Relations If we denote the ordered pairs by (x, y), then the set of all x  values is the DOMAIN and the set of all y  values is the RANGE. A relation is a set of ordered pairs.

6. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 6 Example: Finding the domain and range of a relation A physics class measured the time y that it takes for an object to fall x feet, as shown in the table. The object was dropped twice from each height. a. Express the data as a relation S. b. Find the domain and range of S. Domain = {20, 40} Range = {1.1, 1.2, 1.5, 1.6} x (feet)20 40 y (seconds)1.21.11.51.6

7. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 7 Cartesian (rectangular) coordinate plane Also referred to as the xy-plane. Horizontal axis is called the x-axis. Vertical axis is called the y-axis. Axes intersect at the origin. Axes determine four regions called quadrants, numbered I, II, III, IV, counterclockwise. The term scatterplot is given to a graph in the xy- plane where distinct points are plotted.

8. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 8 Cartesian (rectangular) coordinate plane Here’s an example of a scatterplot.

9. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 9 Example: Graphing a Relation Complete the following for the relation S = {(5, 10), (5, –5), (  10, 10), (0, 15), (  15, 10)} (a)Find the domain and range of this relation. (b)Determine the maximum and minimum of the x-values and then of the y-values. (c)Label appropriate scales on the xy-axes. (d)Plot the relation.

10. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 10 Example: Graphing a Relation S = {(5, 10), (5, –5), (  10, 10), (0, 15), (  15, 10)} (a)Domain: first number in each ordered pair: D = {–15, –10, 0, 5} Range: second number in each ordered pair: R = {–10, –5, 10, 15} (b) x-minimum: –15; x-maximum: 5 y-minimum: –10; y-maximum: 15 (c)An appropriate scale for both the x and y-axes is –20 to 20, with each tick mark representing 5.

11. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 11 Example: Graphing a Relation (d)Plot the relation.

12. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 12 Line Graph Sometimes it is helpful to connect consecutive data points in a scatterplot with straight-line segments. This type of graph, which visually emphasizes changes in the data, is called a line graph.

13. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 13 Use the table to make a scatterplot of average monthly precipitation in Portland, Oregon. Then make a line graph. Example: Making a scatterplot and a line graph

14. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 14 Example: Making a scatterplot and a line graph

15. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 15 Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) in the xy-plane is

16. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 16 Find the exact distance between (3, –4) and (–2, 7). Then approximate this distance to the nearest hundredth. Solution Example: Finding the distance between two points

17. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 17 Midpoint Formula The midpoint of the segment with endpoints (x 1, y 1 ) and (x 2, y 2 ) in the xy-plane is

18. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 18 Example: Finding the midpoint Find the midpoint of the line segment connecting the points (6, –7) and (–4, 6). Solution

19. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 19 Standard Equation of a Circle The circle with center (h, k) and radius r has equation Note: If the circle is centered at (0, 0), the equation simplifies to

20. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 20 Find the center and radius of the circle with the given equation. Graph each circle. Solution Center: (0, 0) Radius: 3 Example: Finding the center and radius of a circle

21. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 21 Solution (continued) Center: (1, –2) Radius: 2 Example: Finding the center and radius of a circle

22. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 22 Find the equation of the circle that satisfies the conditions. Graph each circle. (a) Radius 4, center (–3, 5) (b) Center (6, –3) with the point (1, 2) on the circle. a. Let r = 4 and (h, k) = (–3, 5) Solution Example: Finding the equation of a circle

23. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 23 (b) Center (6, –3) with the point (1, 2) on the circle. Find the distance between the center and point on the circle. Example: Finding the equation of a circle