Mathematics Foundation for College. Lesson Number 8a. Lesson Number 8a Page 1

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1 Mathematics Foundation for College Lesson Number 8a Lesson Number 8a Page 1

2 Lesson Number 8 Topics to be Covered in this Lesson Coordinate graphing, linear equations, conic sections. Lesson Number 8a Page 2

3 René Descartes René Descartes was a famous French philosopher, scientist and mathematician who lived in the years from 1596 to The most comprehensive of Descartes' works, known as Principia Philosophiae, was published in Amsterdam in Descartes also published treatises on physics, meteorology and mathematics, including one titled La Géométrie. In this treatise, Descartes developed a creative and innovative theory that linked algebra and geometry together in what became known as the Cartesian system. Lesson Number 8a Page 3

4 Cartesian System The Cartesian system that was invented by René Descartes, and that was named in honor of him, became the foundation for the areas of mathematics known today as calculus and analytic geometry. The Cartesian system allows phenomena in the real world to be depicted in the form of two dimensional and three dimensional graphs that can be expressed as algebraic equations. A simple form of expression in the Cartesian system would be a linear equation that results in the depiction of a straight line on a two dimensional graph. Lesson Number 8a Page 4

5 Height of an Elevator Suppose that an elevator rises at a rate of two yards per second. Assume that the elevator was four yards below ground level three seconds ago. Let Y represent the height of the elevator below or above ground level at time X: X etc. Y etc. The following equation expresses the linear relationship between X and Y: Y = 2 X + 2 The following graph depicts this relationship. Lesson Number 8a Page 5

6 Y ( 1, 4 ) ( -1, 0 ) -X ( 0, 2 ) X ( -3, -4 ) -Y Lesson Number 8a Page 6

7 Slope of a Line The slope of a line is a measure of its upward or downward direction. In the line given by the equation: Y = 2 X + 2 The value of Y increases by 2 for each successive increase in the value of X. So, the slope of this line is equal to 2. This can be inferred from the fact that the coefficient of X is 2. The larger the slope, the higher the upward angle of the line. The smaller the slope, the lower the upward angle of the line. Lesson Number 8a Page 7

8 Y ( 1, 4 ) ( -1, 0 ) -X ( 0, 2 ) X ( -3, -4 ) -Y Lesson Number 8a Page 8

9 Line Through the Origin A line that passes through the origin, that is, the center of the coordinate system, is a special case in which the X-intercept and the Y-intercept both have a value of zero. The following linear equation would be an example of this: Y = 2 X The graph of this line would be as follows. Lesson Number 8a Page 9

10 Y ( 2, 4 ) ( 0, 0 ) ( 1, 2 ) -X X ( -2, -4 ) -Y Lesson Number 8a Page 10

11 Parallel Lines Two lines are parallel to each other if and only if they have the same slope. Therefore, the following linear equations: Y = 2 X + 2 Y = 2 X must represent parallel lines. The following graph of these two lines demonstrates this. Lesson Number 8a Page 11

12 Y Y = 2 X -X X Y = 2 X + 2 -Y Lesson Number 8a Page 12

13 Slope of a Line This linear equation has a slope of 1/2: Y = ( 1/2 ) X + 2 It corresponds to a line with a flatter upward angle. The value of Y increases by 1/2 for each successive increase in the value of X. The following is the graph of this line. Lesson Number 8a Page 13

14 ( -2, 1 ) ( -3, 0.5 ) -X ( -4, 0 ) Y ( 4, 4 ) ( 1, 2.5 ) X ( 0, 2 ) -Y Lesson Number 8a Page 14

15 Negative Slope If the slope of a line is negative, the line slopes downward from left to right rather than upward. For example, in the line given by the equation: Y = -2 X + 2 This line has a slope of -2, which means that the value of Y decreases by 2 for each successive increase in the value of X. The following is the graph of this line. Lesson Number 8a Page 15

16 Y ( -1, 4 ) ( 1, 0 ) -X ( 0, 2 ) X ( 3, -4 ) -Y Lesson Number 8a Page 16

17 Zero Slope If the slope of a line is positive, the line inclines upward from left to right. If the slope of a line is negative, the line declines downward from left to right. If the slope of a line is zero, the line is horizontal. An example of this is the line given by the equation: That is: Y = 0 X + 2 Y = 2 This line has a slope of zero, which means that the Y- coordinate remains constant regardless of the value of the X-coordinate. The following is the graph of this line. Lesson Number 8a Page 17

18 Y ( -3, 2 ) ( 0, 2 ) Y = 2 -X X -Y Lesson Number 8a Page 18

19 No Slope A vertical line has no slope. For example, in the line given by the equation: X = -3 This line has no slope. On this line, the X-coordinate remains constant regardless of the value of the Y- coordinate. The following is the graph of this line. Lesson Number 8a Page 19

20 X = -3 Y ( -3, 2 ) -X X ( -3, -1 ) -Y Lesson Number 8a Page 20

21 Perpendicular Lines If two lines are perpendicular to each other, that is, if the two lines intersect with a 90 degree angle, then the product of the slopes of the two lines is equal to -1. For example, consider the following two linear equations: Y = -2 X + 2 Y = ( 1/2 ) X + 2 The product of the slopes of these two lines is: ( -2 ) x ( 1/2 ) = -1 So, these two lines must be perpendicular. This can be seen on the following graph. Lesson Number 8a Page 21

22 Y = ( 1/2 ) X + 2 Y -X X Y = -2 X + 2 -Y Lesson Number 8a Page 22

23 General Linear Equation Any equation of the following form: a X + b Y = c where a, b and c are constants, is known as a linear equation because its graph is a straight line. Such an equation can be transformed by algebraic manipulation into the following form: Y = m X + n Slope of the line. Y-intercept of the line. Lesson Number 8a Page 23

24 General Linear Equation For example, this linear equation: 6 X - 2 Y = 3 can be transformed into: Y = 3 X - ( 3/2 ) Slope of the line. Y-intercept of the line. The value of Y increases by 3 for each successive increase in the value of X. The line intercepts the Y- axis at (-3/2) that is at (-1.5). The graph of this line is as follows. Lesson Number 8a Page 24

25 Y ( 1, 1.5 ) -X X ( 0, -1.5 ) -Y Lesson Number 8a Page 25

26 Deriving a Linear Equation Thus far, we have been examining the graph of line that is derived from a linear equation. Let s work in the opposite direction by starting with a line and trying to figure out the linear equation that corresponds to the line. Given the coordinates of any two points on a Cartesian system, a line can be drawn through those two points. So, the question to be examined would be how to derive the linear equation that corresponds to that line. For example, suppose we imagine a line passing through the following two points: ( 2, 1 ) and ( 3, 4 ) The graph of the line would be as follows. Lesson Number 8a Page 26

27 Y ( 3, 4 ) ( 2, 1 ) 4-1 = 3 -X X 3-2 = 1 -Y Lesson Number 8a Page 27

28 Deriving a Linear Equation The slope of the line can be determined by calculating the dimensions of the triangle shaded in gray color on this graph. The slope is equal to the height of the triangle divided by the width of the triangle. The height of the triangle is the difference between the Y- coordinates of the two points. The width of the triangle is the difference between the X-coordinates of the two points. So, the slope of this line is: ( Y ( X 1 1 Y2 ) X ) 2 (4 (3 1) 2) That is, the Y-coordinate increases by 3 for each increase by one in the value of the X-coordinate Lesson Number 8a Page 28

29 Deriving a Linear Equation The line will have an equation of the following form: Y = m X + n where m is the slope and n is the y-intercept. Since we know the slope of this line is 3, the equation of the line is of the following form: Y = 3 X + n Next, we have to determine the value of the y-intercept. Lesson Number 8a Page 29

30 Deriving a Linear Equation We could use either of the points ( 2, 1 ) or ( 3, 4 ) to determine the value of the y-intercept. We will use the point ( 2, 1 ). That is, Y = 1 when X = 2. So, the following must be true: Y = 3 X + n 1 = 3 ( 2 ) + n 1 = 6 + n n = - 5 (this is the y-intercept) Therefore, the equation of the line is: Y = 3 X - 5 Lesson Number 8a Page 30

31 Length of Line Segment The length of a line segment between two points can be determined by computing the length of the hypotenuse of the triangle formed by the two points as shown in the following diagram using these points as an example: ( 1, 0 ) and ( 3, 5 ) The length of the hypotenuse of a right triangle is given by the square root of the sum of the squares of the two sides of the triangle. So, the distance between these two points is: (3 1) 2 (5 0) 2 (2) 2 (5) Lesson Number 8a Page 31

32 Y ( 3, 5 ) ( 1, 0 ) 5-0 = 5 -X X 3-1 = 2 -Y Lesson Number 8a Page 32

33 Length of Line Segment The length of a line segment between any two given points ( X 1, Y 1 ) and ( X 2, Y 2 ) is determined as follows: ( X Y X 2) ( Y1 2) This formula works for positive, negative or zero coordinates. Take care to do the arithmetic correctly when dealing with negative coordinates. Lesson Number 8a Page 33

34 Midpoint of Line Segment The midpoint of a line segment can be determined by taking half the distance between the X-coordinates and half the distance between the Y- coordinates as shown in the following graph using the two points: ( 1, 0 ) and ( 3, 5 ) The X-coordinate of the mid-point of the segment is: (1 3) The Y-coordinate of the mid-point of the segment is: (0 5) Lesson Number 8a Page 34

35 Y ( 3, 5 ) ( 1, 0 ) -X Mid- Point ( 2, 2.5 ) X -Y Lesson Number 8a Page 35

36 Midpoint of Line Segment The mid-point of a line segment between any two points ( X 1, Y 1 ) and ( X 2, Y 2 ) is determined as follows. The X-coordinate of the mid-point of the segment is: ( X 1 X 2) 2 The Y-coordinate of the mid-point of the segment is: ( Y 1 Y2) 2 Either formula works for positive, negative or zero coordinates. Lesson Number 8a Page 36