L13-Mon-3-Oct-2016-Sec-1-1-Dist-Midpt-HW Graph-HW12-Moodle-Q11, page 1 L13-Mon-3-Oct-2016-Sec-1-1-Dist-Midpt-HW Graph-HW12-Moodle-Q11

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1 L13-Mon-3-Oct-016-Sec-1-1-Dist-Midpt-HW11-1--Graph-HW1-Moodle-Q11, page 1 L13-Mon-3-Oct-016-Sec-1-1-Dist-Midpt-HW11-1--Graph-HW1-Moodle-Q Rectangular Coordinate System: Suppose we know the sum of two numbers is 6. We could represent this situation with the equation x + y = 6. There are an infinite number of possibilities for x and y: 1 and 5; -4 and 10; 0 and 6; etc. We could visualize this relation between x and y by plotting points using two number lines, one for x and the other for y.

2 L13-Mon-3-Oct-016-Sec-1-1-Dist-Midpt-HW11-1--Graph-HW1-Moodle-Q11, page A better way is to do this: Draw a horizontal number line. This is usually called the x-axis and is labeled with the variable x. Next, draw a vertical number line perpendicular to the x-axis with the two number lines intersecting at zero. The vertical number line is usually called the y-axis and is labeled with the variable y. The point of intersection, the zero of each number line, is called the origin.

3 L13-Mon-3-Oct-016-Sec-1-1-Dist-Midpt-HW11-1--Graph-HW1-Moodle-Q11, page 3 Now, plot points where the x and y coordinates satisfy the equation x + y = 6: The flat surface that the axes and the grid lie on is called the xy-plane.

4 L13-Mon-3-Oct-016-Sec-1-1-Dist-Midpt-HW11-1--Graph-HW1-Moodle-Q11, page 4 Taken together, the number lines and grid, form a rectangular coordinate system. Because we typically use x and y for the variables, a rectangular coordinate system is often called an xycoordinate system. Also, because the French mathematician Rene Descartes is credited with developing this type of coordinate system, it is also referred to as the Cartesian coordinate system. Every point in the xy-plane has two numbers associated with it. The x-coordinate or abscissa tells how far the point lies to the left or right of the y-axis. The y-coordinate or ordinate tells how far the point lies above or below the x-axis. The x-and y-coordinate are often written inside parentheses like this: (x, y). The first number always represents the x-coordinate and the second number always represents the y-coordinate. Because the order in which the numbers is written is important, (x, y) is called an ordered pair. The x- and y-axes divide the plane into four regions called quadrants. These are labeled with the Roman numerals I, II, III, and IV in a counter-clockwise direction beginning in the upper right. Points on the axes do not lie in a quadrant. The rectangular coordinate system can be used to connect algebra and geometry. For example, suppose we have any two points x y and x, y. Note use of subscripts. We want to find the distance between the points. Use Pythagoras to find the hypotenuse., This gives us the Distance Formula (memorize this): d P, P x x y y

5 L13-Mon-3-Oct-016-Sec-1-1-Dist-Midpt-HW11-1--Graph-HW1-Moodle-Q11, page 5 It looks like there will be two points. If the points are on the y-axis then their x component must be 0. So, we have the two points (4, 4) and (0, y) that are a distance of 5 units from each other., d P P x x y y y y y y 3 4 y or 3 4 y y 1 or y 7 So, the two points are (0, 1) and (0, 7).

6 L13-Mon-3-Oct-016-Sec-1-1-Dist-Midpt-HW11-1--Graph-HW1-Moodle-Q11, page 6 The midpoint x, y of a line segment connecting points P 1 x, y 1 1 and P x, y is the point where the distance from x, y 1 1 to x, y is the same as the distance from x, y to x, y. We can use the distance formula to find x, y. x y x y x y x y 1, 1, x x 1 y y1 x x y y x x y y x x y y distance from to, distance from to, 1 1 The x and y variables are completely independent of each other. That is, for a given x, y can be anything and vice versa. So, we can break up this equation into two equations, one involving x and the other involving y. Then, we can solve each equation independently. x x x x 1 x x x x 1 x x x x or x x x x 1 1 x x or x x x x 1 1 x x x 1 x x 1 x x1 = x means the two points are the same so we ignore that solution. x x 1 The y part works the same. This gives the formulas: x midpoint y y 1 y midpoint

7 L13-Mon-3-Oct-016-Sec-1-1-Dist-Midpt-HW11-1--Graph-HW1-Moodle-Q11, page 7

8 L13-Mon-3-Oct-016-Sec-1-1-Dist-Midpt-HW11-1--Graph-HW1-Moodle-Q11, page 8 Let d = distance from balloon to intersection (feet) Let t = time since balloon passed intersection (seconds) From the diagram we see that we have a right triangle and want to find the hypotenuse. So, use Pythagoras: d dist along road vertical dist Hypotenuse d feet Vertical Leg 100 feet 15 miles Horizontal Leg 15 mpht t hours 15t miles 1 hour Since we use feet and seconds we must convert 15 mph to feet per second before using Pythagoras. 15 mi 580 ft 1 hr 15mph 1 hr 1 mi 3600 sec feet 1 second ft per sec

9 L13-Mon-3-Oct-016-Sec-1-1-Dist-Midpt-HW11-1--Graph-HW1-Moodle-Q11, page 9 d dist along road vertical dist ft t sec 1sec t ,000 Here is the graph: 100 ft Notice that when t gets very large, 484t completely dominates the 10,000 and we have d for t 484t 10,000 d 484t t That is lim 484t 10,000 t which is a straight line. Check: When time t 0, d ,000 10, ft t

10 L13-Mon-3-Oct-016-Sec-1-1-Dist-Midpt-HW11-1--Graph-HW1-Moodle-Q11, page 10 x-intercept: Value of x when y = 0: 0 x x 6 x x 0 3 x 3 or x x -int: 3,0 and,0 y-intercept: Value of y when x = 0 y y y -int: 0, 6

11 L13-Mon-3-Oct-016-Sec-1-1-Dist-Midpt-HW11-1--Graph-HW1-Moodle-Q11, page 11 x-intercept: x 0 16 y-intercept: 0 x 1 x y ,0 and 3.5,0 y 16 y 4 or y 4 y 6 or y 0,6 or 0,

12 L13-Mon-3-Oct-016-Sec-1-1-Dist-Midpt-HW11-1--Graph-HW1-Moodle-Q11, page 1

13 L13-Mon-3-Oct-016-Sec-1-1-Dist-Midpt-HW11-1--Graph-HW1-Moodle-Q11, page 13 We can test for symmetries if we have the equation by doing the following: Symmetry about x axis: Replace y with -y and simplify. If you get the original equation you have symmetry about the x axis. Symmetry about y axis: Replace x with -x and simplify. If you get the original equation you have symmetry about the y axis. Using function notation: f x f x Symmetry about origin: Replace x with -x and simplify. If you get the NEGATIVE of the original equation you have symmetry about the origin. Using function notation: f x f x Say whether each of these has x, y, or origin symmetry or none of these. 5 y x x x x y 5 x 5 x x Since we get back -y, this is symmetric about the origin.

14 L13-Mon-3-Oct-016-Sec-1-1-Dist-Midpt-HW11-1--Graph-HW1-Moodle-Q11, page 14 y x 4 x x x x 1 4 x 1 y Since we get back y, this is symmetric about the y-axis. x 4 y x x x x 4 x 4 x 4 y x x x Since we get back -y, this is symmetric about the origin.

15 L13-Mon-3-Oct-016-Sec-1-1-Dist-Midpt-HW11-1--Graph-HW1-Moodle-Q11, page 15 Complete the graph to make it Symmetric about the x-axis Complete the graph to make it Symmetric about the y-axis Complete the graph to make it Symmetric about the y-axis