The Coordinate System and Graphs
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1 The Coordinate System and Graphs Fall Math 1010 (Math 1010) M / 17
2 Roadmap Plotting ordered pairs. The distance formula. The midpoint formula. Graphs of equations. Intercepts. Verifying solutions to equations. (Math 1010) M / 17
3 3.1 - Plotting Points 3 2 A B 1 C 0 D E -1 F -2-3 Point B is the ordered pair (-2,1). What are the coordinates of each point? (Math 1010) M / 17
4 3.1 - Plotting Points Previous slide: Point Coordintes A (3, 2) B ( 3, 1) C (0, 0) D (3, 0) E (2.6, 1) F ( 1, 2) (Math 1010) M / 17
5 3.1 - Plotting Points To plot a point, the ordered pair, (x, y) describes the coordiante by moving x units along the horizontal axis, and y units along the vertical axis. (Math 1010) M / 17
6 3.1 - Plotting Points To plot a point, the ordered pair, (x, y) describes the coordiante by moving x units along the horizontal axis, and y units along the vertical axis. Tips: Plot with a solid dot. Give the point a clean label. (Math 1010) M / 17
7 3.1 - Plotting Points To plot a point, the ordered pair, (x, y) describes the coordiante by moving x units along the horizontal axis, and y units along the vertical axis. Tips: Plot with a solid dot. Give the point a clean label. An ordered pair can lie in one of four quadrants, or on the border. (Math 1010) M / 17
8 3.1 - Plotting Points (Math 1010) M / 17
9 3.1 - Plotting Points Any problems with this plot? (Math 1010) M / 17
10 3.1 - Plotting Points 3 Quadrant II 2 Quadrant I A B 1 C 0 D E -1 F -2 Quadrant III -3 Quadrant IV Labels return. Which quadrant is each point within? (Quadrants are labelled for convenience.) (Math 1010) M / 17
11 3.1 - Distance Formula Distance between two reals is the absolute value of the distance. distance = b a (Math 1010) M / 17
12 3.1 - Distance Formula Distance between two reals is the absolute value of the distance. From the Pythagorean Theorem, distance = b a The sum of the squares of the legs of a right triangle equals the square of the hypotenuse. (Math 1010) M / 17
13 3.1 - Distance Formula Distance between two reals is the absolute value of the distance. From the Pythagorean Theorem, distance = b a The sum of the squares of the legs of a right triangle equals the square of the hypotenuse. The distance between two points in the Cartesian coordinate frame, (x 1, y 1 ) and (x 2, y 2 ) is distance = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 (Math 1010) M / 17
14 3.1 - Distance Formula 5 4 (2,4) 3 (-1,2) Distance? This is the hypotenuse of some right triangle. (Math 1010) M / 17
15 3.1 - Distance Formula (2,4) 2 (-1,2) = 13. The distance is 13. Also, (2 ( 1)) 2 + (4 2) 2 = 13. (Math 1010) M / 17
16 3.1 - Midpoint Two points make a line segment. The middle of that segment is the midpoint. (Math 1010) M / 17
17 3.1 - Midpoint Two points make a line segment. The middle of that segment is the midpoint. Example: The points ( 5, 3) and (9, 3) can be connected by a line segment. The midpoint is ( ) ( 5) ( 3) = ( ) = (2, 0) 2 (Math 1010) M / 17
18 3.1 - Midpoint Two points make a line segment. The middle of that segment is the midpoint. Example: The points ( 5, 3) and (9, 3) can be connected by a line segment. The midpoint is ( ) ( 5) ( 3) = ( ) = (2, 0) 2 The Midpoint Formula between (x 1, y 1 ) and (x 2, y 2 ) is ( x1 + x 2 Midpoint =, y ) 1 + y (Math 1010) M / 17
19 3.2 - Graphs of Functions How would you graph all points that are the same distance from (0, 0) and (3, 2)? (Math 1010) M / 17
20 3.2 - Graphs: Point-Plotting Method of Sketching A Graph 1. (Ideal) Rewrite the equation by isolating one of the variables. 2. Make a table of values showing several solution points. 3. Plot these ordered pairs on a system with a reasonable scale. 4. Smoothly connect the points. (Sharp turns are graphed from experience.) Example: Plot 4x y = 1. (Math 1010) M / 17
21 3.2 - Graphs: Point-Plotting Method of Sketching A Graph 1. (Ideal) Rewrite the equation by isolating one of the variables. 2. Make a table of values showing several solution points. 3. Plot these ordered pairs on a system with a reasonable scale. 4. Smoothly connect the points. (Sharp turns are graphed from experience.) Example: Plot 4x y = 1. (Math 1010) M / 17
22 3.2 - Absolute Value Equations Absolute value equations are sharp at one or more places. Example: Use the point-plotting method on y = x. Choose values for x from { 3, 2,..., 2, 3}. (Math 1010) M / 17
23 3.2 - Absolute Value Equations Absolute value equations are sharp at one or more places. Example: Use the point-plotting method on y = x. Choose values for x from { 3, 2,..., 2, 3}. x y Solution point (-3,3) (-2,2) (-1,1) (0,0) (1,1) (2,2) (3,3) (Math 1010) M / 17
24 3.2 - Absolute Value Equations Absolute value equations are sharp at one or more places. Example: Use the point-plotting method on y = x. Choose values for x from { 3, 2,..., 2, 3}. x y Solution point (-3,3) (-2,2) (-1,1) (0,0) (1,1) (2,2) (3,3) For the plot, do any patterns emerge? What point is the sharp corner? (Math 1010) M / 17
25 3.2 - Intercepts Intercepts are points on a graph that also intersects an axis. An x intercept has the corrdinate (a, 0), and a y intercept has the coordinate (0, b). That is, either the y or the x variable is zero, and the other is a value that solves the original equation. (Math 1010) M / 17
26 3.2 - Intercepts Find the intercepts of y = 4x 8. (Math 1010) M / 17
27 3.2 - Intercepts Find the intercepts of y = 4x 8. x: Let y = 0. Then, This is ( 2, 0). 0 = 4x 8 8 = 4x x = 2 (Math 1010) M / 17
28 3.2 - Intercepts Find the intercepts of y = 4x 8. x: Let y = 0. Then, This is ( 2, 0). y: Let x = 0. Then, This is (0, 8). 0 = 4x 8 8 = 4x x = 2 y = 0 8 y = 8 (Math 1010) M / 17
29 Assignment Assignment: For Wednesday: 1. Exercises from 3.1, 3.2 due Wednesday, September Pre-Exam #1: September 18. Chapters 1 & 2 3. Read section 3.3. (Math 1010) M / 17
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