MAT 125 – Applied Calculus 1.4 Straight Lines

Today’s Class  We will be learning the following concepts in Section 1.3:  The Cartesian Coordinate System  The Distance Formula  The Equation of a Circle  We will be learning the following concepts in Section 1.4:  Slope of a Line  Equations of Lines Dr. Erickson 1.4 Straight Lines 2

Slope of a Nonvertical Line  If (x 1, y 1 ) and (x 2, y 2 ) are two distinct points on a nonvertical line L, then the slope m of L is given by (x 1, y 1 ) (x 2, y 2 ) y x L y 2 – y 1 =  y x 2 – x 1 =  x Dr. Erickson 1.4 Straight Lines 3

Slope of a Vertical Line  Let L denote the unique straight line that passes through the two distinct points (x 1, y 1 ) and (x 2, y 2 ). If x 1 = x 2, then L is a vertical line, and the slope is undefined. (x 1, y 1 ) (x 2, y 2 ) y x L Dr. Erickson 1.4 Straight Lines 4

Slope of a Nonvertical Line  If m > 0, the line slants upward from left to right. y x L  y = 2  x = 1 m = 2 Dr. Erickson 1.4 Straight Lines 5

Slope of a Nonvertical Line  If m < 0, the line slants downward from left to right. m = –1 y x L  y = –1  x = 1 Dr. Erickson 1.4 Straight Lines 6

 Sketch the straight line that passes through the point (1, 2) and has slope – y x Example 1 Dr. Erickson 1.4 Straight Lines 7

 Find the slope m of the line that goes through the points (–2, –2) and (4, –4). Example 2 Dr. Erickson 1.4 Straight Lines 8

 Let L be a straight line parallel to the y-axis. Then L crosses the x- axis at some point (a, 0), with the x-coordinate given by x = a, where a is a real number. Any other point on L has the form (a, y), where y is an appropriate number. The vertical line L can therefore be described as x = a Equations of Lines (a, y ) y x L (a, 0) Dr. Erickson 1.4 Straight Lines 9

Equations of Lines  Let L be a nonvertical line with a slope m.  Let (x 1, y 1 ) be a fixed point lying on L and (x, y) be variable point on L distinct from (x 1, y 1 ).  Using the slope formula by letting (x 2, y 2 ) = (x, y) we get  Multiplying both sides by x – x 1 we get Dr. Erickson 1.4 Straight Lines 10

Point-Slope Form of an Equation of a Line An equation of the line that has slope m and passes through point (x 1, y 1 ) is given by Dr. Erickson 1.4 Straight Lines 11

 Find an equation of the line that passes through the point (2, 4) and has slope –1. Example 5 Dr. Erickson 1.4 Straight Lines 12

 Find an equation of the line that passes through the points (–1, –2) and (3, –4). Example 6 Dr. Erickson 1.4 Straight Lines 13

Parallel Lines  Two distinct lines are parallel if and only if their slopes are equal or their slopes are undefined. Dr. Erickson 1.4 Straight Lines 14

Perpendicular Lines  If L 1 and L 2 are two distinct nonvertical lines that have slopes m 1 and m 2, respectively, then L 1 is perpendicular to L 2 (written L 1 ┴ L 2 ) if and only if Dr. Erickson 1.4 Straight Lines 15

Example 7  Find an equation of the line that passes through the point (2, 4) and is perpendicular to the line  Find an equation of the line that passes through the origin and is parallel to the line joining the points (2,4) and (4,7). Dr. Erickson 1.4 Straight Lines 16

Crossing the Axis  A straight line L that is neither horizontal nor vertical cuts the x-axis and the y-axis at, say, points (a, 0) and (0, b), respectively.  The numbers a and b are called the x-intercept and y-intercept, respectively, of L. (a, 0) (0, b) y x L y-intercept x-intercept Dr. Erickson 1.4 Straight Lines 17

Slope Intercept Form of an Equation of a Line  An equation of the line that has slope m and intersects the y-axis at the point (0, b) is given by y = mx + b Dr. Erickson 1.4 Straight Lines 18

Example 8  Find the equation of the line that has the following: m = –1/2; b = 3/4 Dr. Erickson 1.4 Straight Lines 19

Example 9  Determine the slope and y-intercept of the line whose equation is 3x – 4y + 8=0. Dr. Erickson 1.4 Straight Lines 20

Example 10 Dr. Erickson 1.4 Straight Lines 21

General Form of an Linear Equation The equation Ax + By + C = 0 where A, B, and C are constants and A and B are not both zero, is called the general form of a linear equation in the variables x and y. Dr. Erickson 1.4 Straight Lines 22

Theorem 1  An equation of a straight line is a linear equation; conversely, every linear equation represents a straight line. Dr. Erickson 1.4 Straight Lines 23

Example 11  Sketch the straight line represented by the equation 3x – 2y +6 = 0. Dr. Erickson 1.4 Straight Lines 24

Next Class  We will discuss the following concepts:  Functions  Determining the Domain of a Function  Graphs of Functions  The Vertical Line Test  Please read through Section 2.1 – Functions and Their Graphs in your text book before next class. Dr. Erickson 1.4 Straight Lines 25