Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. 1.2 Straight Lines Slope Point-Slope Form Slope-Intercept Form General Form

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

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. The number is a measure of the vertical change in y, and is a measure of the horizontal change in x. The slope m is a measure of the rate of change of y with respect to x. If and are two distinct points on a nonvertical line L, then the slope m of L is given by Slope of a Nonvertical Line

Examples Find the slope m of the line that goes through the points (4, 1) and (6, -3). Solution Choose (x 1, y 1 ) to be (4, 1) and (x 2, y 2 ) to be (6, -3). With x 1 = 4, y 1 = 1, x 2 = 6, y 2 = -3, we find

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Two lines are parallel if and only if their slopes are equal or both undefined Two lines are perpendicular if and only if the product of their slopes is –1. That is, one slope is the negative reciprocal of the other slope (ex. ). Parallel Lines and Perpendicular Lines

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Equations of Lines Let L be a straight line parallel to the y-axis. Then the vertical line L is described by the sole condition x = a. Next, suppose L is a nonvertical line so that it has a well-defined slope m. Suppose (x 1, y 1 ) is a fixed point lying on L and (x, y) is a variable point on L distinct from (x 1, y 1 ). Then the slope is given by

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Point-Slope Form An equation of a line that passes through the point with slope m is given by: Ex. Find an equation of the line that passes through (3,1) and has slope m = 4.

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. A straight line L that is neither horizontal nor vertical cuts the x-axis an the y-axis at points (a, 0) and (0, b) respectively. The numbers a and b are called the x-intercept and y- intercept, respectively. Now let L be a line with slope m an y-intercept. With the point given by (0, b) and slope m, we have x-intercept and y-intercept

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Slope-Intercept Form An equation of a line with slope m and y-intercept is given by: Ex. Find an equation of the line that passes through (0,-4) and has slope.

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Example Determine the slope and y-intercept of the line whose equation is Solution: Step 1. Step 2.

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Example Find an equation of the line that passes through (-2, 1) and is perpendicular to the line Solution: Step 1. Step 2.

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. General Form The general form of an equation of a line is given by: Where A, B, and C are constants and A and B are not both zero. *Note: An equation of a straight line is a linear equation and every linear equation represents a straight line.

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Vertical Lines x = 3 Can be expressed in the form x = a x y

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Horizontal Lines y = 2 Can be expressed in the form y = b x y

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. y-axis (–2, 3) Ex. Find an equation of the line that passes through the point (–2, 3) and is parallel to the y – axis. Vertical Line: x = –2 x y

Applied Example Suppose an art object purchased for $50,000 is expected to appreciate in value at a constant rate of $5000 per year for the next 5 years. Write an equation predicting the value of the art object for any given year. What will be its value 3 years after the purchase? Applied Example 11, page 16

Solution Let x = time (in years) since the object was purchased y = value of object (in dollars) Then, y = 50,000 when x = 0, so the y-intercept is b = 50,000. Every year the value rises by 5000, so the slope is m = Thus, the equation must be y = 5000x + 50,000. After 3 years the value of the object will be $65,000: y = 5000(3) + 50,000 = 65,000