Section 2.1: Rectangular Coordinates, Distance, Midpoint Formulas I. Graphs of Equations The linking of algebra and geometry:

Transcription

1 Section 2.1: Rectangular Coordinates, Distance, Midpoint Formulas I. Graphs of Equations The linking of algebra and geometry: 1 (René Descartes ) Rectangular Cartesian Coordinate System Terms: x-axis; y-axis; origin (0,0); Quadrants I IV; xy-plane A point in the plane is assigned an ordered pair (a,b); a is the x-coordinate, b is the y-coordinate. Example: Find the distance between the points P= (2, 3) and P 2 = ( 3,2). 1 Distance Formula (Page 176): The distance between two points P 1 and P is given by 2 d( P, P ) = ( x x ) + ( y y )

2 Section 2.1: Cartesian Coordinates (continued) Example: Find the midpoint of the line segment between the points P= 1 (2, 3) and P 2 = ( 3,2). Midpoint Formula (Page 66): The midpoint of the line segment from P 1 to P 2 is x + x y + y ( xm, ym), 2 2 Page = 2

3 Section 2.2: Graphing Equations, Intercepts, Symmetry The graph of an equation in two variables x and y is the set of points in the xy-plane whose coordinates ( x, y ) make the equation true. A complete graph presents enough of the graph so that a viewer of the illustration will see the rest of the graph as an obvious continuation of what is actually there. Example: Graph y= x+ 2 by plotting points. What is an intercept? Procedure to find Intercepts (p ) 1. An x-intercept has the form ( a,0). To find the x- intercept(s), if any, let y = 0 in the equation and solve for x. 2. An y-intercept has the form (0, b ). To find the y- intercept(s), if any, let x = 0 in the equation and solve for y. Example: Algebraically find the intercepts of y= x+ 2. Plot and find the intercepts: y= ( x 1) 2 2. Page

4 For your WileyPlus homework for this section: When you input an expression for a line in WileyPlus, you must use * to indicate multiplication, e.g., y = 3/5*x 8. In everyday conversation, the term line is used ambiguously. In mathematics, a line requires two points. A line is the shortest distance between two points and extends indefinitely in both directions. To approximate real life situations we use a model. A frequently used mathematical model is a linear model. Example: Graph these on the same coordinate system: y= 2x+ 0; y= 2x+ 3; y= 2x 3. - These lines are parallel they have the same slope, x coefficient. - The y-intercept is the number (constant) terms. Example: Graph these on the same coordinate system: y= 0x+ 3; y= 1/2x+ 3; y= x+ 3; y= 2x Line y= 3 is horizontal. - The lines y= mx+ 3 slant upward from left to right, and, as m increases, the line gets steeper. 4

5 Example: Graph these on the same coordinate system by inserting negative sign: y= 0x+ 3; y= 1/2x+ 3; y= x+ 3; y= 2x+ 3 - Line y= 3 is horizontal. - The lines y= mx+ 3 all slant downward, left to right, and, as m decreases (gets more negative), the line gets steeper. The three above examples illustrate the Slope-Intercept Form of a Line (page 199): The formula y= mx+ b is the equation of a line with slope m and y-intercept b. 5

6 Lines are sometimes in general form: Ax+ By= C. By solving for y, an equation in general form can be written in slope-intercept form. Example: Write 2x+ 3y= 6 in slope-intercept form and then graph on your calculator. What is the slope of this line? What is the y-intercept? What is the x-intercept? Example: Choose any pair of points on this line. For example, the points (6, 2) and ( 6,6) lie on this line. ( x, y ) and ( x, y ) rise y y 2 1 run x x (6, 2) and ( 6,6) 2 1 Definition: The slope m of a line between any two points( x 1, y 1 ) and ( x, y ) is given by 2 2 m rise y y = 2 1 y run= x x = x (page 196)

7 Example: We saw that y = 3 is horizontal; x= 2 is vertical. What are the slopes and intercepts of these lines? The equation of a horizontal line is y= b which has slope and intercept(s). The equation of a vertical line is x= a which has slope and intercept(s). Point-Slope Equation of a Line (page 200): The formula y y1 = m( x x1 ) is the equation of a line with slope m and passing through the point ( x1, y 1). In the plane, parallel lines do not intersect. The lines y= 2 x; y= 2x+ 3; y= 2x 3 are parallel. In general, two distinct lines are parallel if and only if they have the same slope. In the plane, perpendicular lines intersect at right (90º) angles. Example: Graph the lines y= 1/4x 3; y= 4x+ 1. Two lines are perpendicular if and only if their slopes are both opposite and reciprocal. 7

8 Examples: Page , 60, 70, 90, 94 WileyPlus example 8