1 / 41

4 th EDITION

College Algebra & Trigonometry and Precalculus. 4 th EDITION. 2.1. Rectangular Coordinates and Graphs. Ordered Pairs The Rectangular Coordinate System The Distance Formula The Midpoint formula Graphing Equations. Ordered Pairs.

todd-hinton
Télécharger la présentation

4 th EDITION

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. College Algebra & Trigonometry and Precalculus 4th EDITION

  2. 2.1 Rectangular Coordinates and Graphs Ordered Pairs The Rectangular Coordinate System The Distance Formula The Midpoint formula Graphing Equations

  3. Ordered Pairs An ordered pair consists of two components, written inside parentheses, in which the order of the components is important.

  4. WRITING ORDERED PAIRS Example 1 Use the table to write ordered pairs to express the relationship between each type of entertainment and the amount spent on it.

  5. WRITING ORDERED PAIRS Example 1 a. DVD rental sales Solution: Use the data in the second row: (DVD rental sales, $71).

  6. WRITING ORDERED PAIRS Example 1 b. Movie tickets Solution: Use the data in the last row: (Movie tickets, $30).

  7. The Rectangular Coordinate System y-axis Quadrant I Quadrant II P(a, b) b x-axis a 0 Quadrant III Quadrant IV

  8. The Distance Formula Using the coordinates of ordered pairs, we can find the distance between any two points in a plane. y Horizontal side of the triangle has length Q(8, 3) P(–4, 3) x R(8, –2) Definition of distance.

  9. The Distance Formula y Vertical side of the triangle has length… Q(8, 3) P(–4, 3) x R(8, –2)

  10. The Distance Formula y By the Pythagorean theorem, the length of the remaining side of the triangle is Q(8, 3) P(–4, 3) x R(8, –2)

  11. The Distance Formula y So the distance between (–4, 3) and (8, –4) is 13. Q(8, 3) P(–4, 3) x R(8, –2)

  12. The Distance Formula y To obtain a general formula for the distance between two points in a coordinate plane, let P(x1, y1) and R(x2, y2) be any two distinct points in a plane. Complete a triangle by locating point Q with coordinates (x2, y1). The Pythagorean theorem gives the distance between P and R as Q(x2, y1) P(x1, y1) x R(x2, y2)

  13. The Distance Formula Note Suppose that P(x1, y1) and R(x2, y2) are two points in a coordinate plane. Then the distance between P and R, written d(P, R) is given by

  14. USING THE DISTANCE FORMULA Example 2 Find the distance between P(–8, 4) and Q(3, –2.) Solution: Be careful when subtracting a negative number.

  15. Using the Distance Formula If the sides a, b, and c of a triangle satisfy a2 + b2 = c2, then the triangle is a right triangle with legs having lengths a and b and hypotenuse having length c?

  16. DETERMINING WHETHER THREE POINTS ARE THE VERTICES OF A RIGHT TRIANGLE Example 3 Are points M(–2, 5), N(12, 3), and Q(10, –11) the vertices of a right triangle. Solution This triangle is a right triangle if the square of the length of the longest side equals the sum of the squares of the lengths of the other two sides.

  17. DETERMINING WHETHER THREE POINTS ARE THE VERTICES OF A RIGHT TRIANGLE Example 3 y N(12, 3) M(–2, 5) x Q(10, –11)

  18. DETERMINING WHETHER THREE POINTS ARE THE VERTICES OF A RIGHT TRIANGLE Example 3 y N(12, 3) M(–2, 5) x Q(10, –11)

  19. DETERMINING WHETHER THREE POINTS ARE THE VERTICES OF A RIGHT TRIANGLE Example 3 y N(12, 3) M(–2, 5) x Q(10, –11)

  20. DETERMINING WHETHER THREE POINTS ARE THE VERTICES OF A RIGHT TRIANGLE Example 3 y N(12, 3) M(–2, 5) x R(10, –11)

  21. Colinear We can tell if three points are collinear, that is, if they lie on a straight line, using a similar procedure Three points are collinear if the sum of the distances between two pairs of points is equal to the distance between the remaining pair of points.

  22. DETERMINING WHETHER THREE POINTS ARE COLLINEAR Example 4 Are the points (–1, 5), (2,–4), and (4, –10) collinear? Solution The distance between (–1, 5) and (2,–4) is

  23. DETERMINING WHETHER THREE POINTS ARE COLLINEAR Example 4 Are the points (–1, 5), (2,–4), and (4, –10) collinear? Solution The distance between (2,–4) and (4,–10) is

  24. DETERMINING WHETHER THREE POINTS ARE COLLINEAR Example 4 Are the points (–1, 5), (2,–4), and (4, –10) collinear? Solution The distance between the remaining pair of points (–1, 5) and (4,–10) is

  25. Midpoint Formula The midpoint of the line segment with endpoints (x1, y1) and (x2, y2) has coordinates

  26. USING THE MIDPOINT FORMULA Example 5 Use the midpoint formula to determine the following. a. Find the coordinates of M of the segment with endpoints (8, –4) and (–6,1). Solution Substitute in the midpoint formula.

  27. USING THE MIDPOINT FORMULA Example 5 Use the midpoint formula to determine the following. b. Find the coordinates of the other endpoint B of a segment with one endpoint A(–6, 12) and midpoint M(8, –2). Solution Let (x, y) represent the coordinates of B. Use the midpoint formula twice.

  28. USING THE MIDPOINT FORMULA Example 5 Let (x, y) represent the coordinates of B. Use the midpoint formula twice. x-value of A x-value of M x-value of A y-value of M Substitute carefully. The coordinates of endpoint B are (22, –16).

  29. APPLYING THE MIDPOINT FORMULA TO DATA Example 6 The graph indicates the increase of McDonald’s restaurants worldwide from 20,000 in 1996 to 31,000 in 2006. Use the midpoint formula and the two given endpoints to estimate the number of restaurants in 2001, and compare it to the actual (rounded) figure of 30,000

  30. APPLYING THE MIDPOINT FORMULA TO DATA Example 6 Number of McDonald’s Restaurants Worldwide (in thousands) 35 30 25 20 2006 1996 Year

  31. APPLYING THE MIDPOINT FORMULA TO DATA Example 6 Solution:2001 is halfway between 1996 and 2006. Find the coordinates of the midpoint of the segment that has endpoints (1996, 20) and (2006, 31). Our estimate is 25,500, which is well below the actual figure of 30,000.

  32. FINDING ORDERED PAIRS THAT ARE SOLUTIONS OF EQUATIONS Example 7 For the equation, find at least three ordered pairs that are solutions a. Solution: Choose any real number for x or y and substitute in the equation to get the corresponding value of the other variable.

  33. FINDING ORDERED PAIRS THAT ARE SOLUTIONS OF EQUATIONS Example 7 a. Solution: Let x = –2. Let y = 3. Multiply. Add 1. Divide by 4. This gives the ordered pairs (–2, –9) and (1, 3). Find one more ordered pair that works!

  34. FINDING ORDERED PAIRS THAT ARE SOLUTIONS OF EQUATIONS Example 7 b. Solution Given equation Let x = 1 Square both sides One ordered pair is (1, 2). Find two more ordered pair!

  35. FINDING ORDERED PAIRS THAT ARE SOLUTIONS OF EQUATIONS Example 7 c. A table provides an organization method for determining ordered pairs. Solution: Five ordered pairs are listed as solutions for this equation.

  36. Graphing an Equation by Point Plotting Step 1 Find the intercepts. Step 2 Find as many additional ordered pairs as needed. Step 3 Plot the ordered pairs from Steps 1 and 2. Step 4 Connect the points from Step 3 with a smooth line or curve.

  37. GRAPHING EQUATIONS Example 8 a. Graph the equation Solution: Step 1 Let y = 0 to find the x-intercept, and let x = 0 to find the y-intercept.

  38. GRAPHING EQUATIONS Example 8 a. Graph the equation Solution: Step 2 We find some other ordered pairs. Let y = 3. Let x = –2. Multiply. Add 1. Divide by 4. This gives the ordered pairs (–2, –9) and (1, 3).

  39. GRAPHING EQUATIONS Example 8 a. Graph the equation Solution: Step 3 Plot the four ordered pairs from Steps 1 and 2. y Step 4 Connect the points with a straight line. x

  40. GRAPHING EQUATIONS Example 8 b. Graph the equation Solution: y Plot the ordered pairs found in example 7b, and develop a fourth point to confirm the direction the curve will take as x increases. x

  41. GRAPHING EQUATIONS Example 8 c. Graph the equation Solution: y x This curve is called a parabola.

More Related