Chapter P: Prerequisite Information

1. Chapter P:Prerequisite Information Section P-2: Cartesian Coordinate System

2. Objectives • You will learn about: • Cartesian plane • Absolute value of a real number • Distance formula • Midpoint formula • Equations of circles • Applications • Why: • These topics provide the foundation for the material that will be covered in this textbook

3. Vocabulary • Cartesian plane (rectangular coordinate system) • X-axis • Y-axis • Origin of the Cartesian plane • Ordered pair (x,y) • Quadrants • Scatter plot • Magnitude • Absolute value of a real number a • Distance between a and b • Pythagorean Theorem • Distance d between points P(x1, y1) and Q(x2, y2) • Midpoint of the line segment with endpoints a and b • Midpoint of the line segment with endpoints (a, b) and (c, d) • Circle • Radius • Center • Standard form of a circle

4. Example 1:Plotting Data on U.S. Exports to Mexico • The value in billions of dollars of U.S. exports to Mexico from 1996 to 2003 is given (on next slide). • Plot the (year, export value) ordered pairs in a rectangular coordinate system.

5. Example 1:Plotting Data on U.S. Exports to Mexico (continued)

6. Example 1:Plotting Data on U.S. Exports to Mexico(continued)

7. Absolute Value of a Real Number • |a|= • a if a > 0 • -a if a < 0 • 0 if a = 0

8. Example 2:Using the Definition of Absolute Value • Evaluate: • |-4| • |p-6|

9. Properties of Absolute Value • Let a and b be real numbers: • |a| ≥ 0 • |a| = |-a| • |ab|= |a||b| • |a/b|= |a|/|b|; b ≠ 0

10. Distance Formula(Number Line) • The distance, d, between a and b is: • |a – b| • Note: |a – b|= |b – a|

11. Distance Formula(Coordinate Plane) • The distance between points P(x1,y1) and Q(x2,y2) is:

12. Example 3:Finding the Distance Between Two Points • Find the distance between the points (1, 5) and (6, 2)

13. Midpoint Formula(Number Line) • The midpoint of the line segment with endpoints a and b is:

14. Example 4:Finding the Midpoint of a Line Segment • Find the midpoint of the line segment with endpoints -9 and 3 on a number line.

15. Midpoint Formula(Coordinate Plane) • The midpoint of a line segment with endpoints (a,b) and (c,d) is:

16. Midpoint Formula(Number Line) • The midpoint of the line segment with endpoints a and b is:

17. Example 5:Finding the Midpoint of a Line Segment • Find the midpoint of the line segment with endpoints (-5, 2) and (3,7)

18. Standard Form Equation of a Circle • The standard form equation of a circle with center (h, k) and radius r is:

19. Example 6:Finding Standard Form Equations of Circles • Finding the standard form equation of the circle with the given information: • Center (-4, 1), radius 8 • Center (0,0), radius 5

20. Example 7:Using an Inequality to Express Distance • We can state that the distance between x and -3 is less than 9. • Write an inequality and solve for x.

21. Example 8:Verifying Right Triangles • Use the converse of the Pythagorean Theorem and the distance formula to show that the points (-3, 4), (1, 0), and (5, 4) determine a right triangle.

22. Example 9:Using the Midpoint Formula • It is a fact from geometry that the diagonals of a parallelogram bisect each other. Prove this with a midpoint formula. • Solution: position a parallelogram in the coordinate plane with the following endpoints: • (0, 0) • (a, b) • (a + c, b) • (c, 0)