Equations of Lines and Linear Models Section 2.4 Equations of Lines and Linear Models

Objectives Point-Slope Form Horizontal and Vertical Lines Parallel and Perpendicular Lines

POINT-SLOPE FORM The line with slope m passing through the point (x1, y1) is given by y = m(x – x1) + y1 or equivalently, y – y1 = m(x – x1), the point-slope form of a line.

Example Find the point-slope form of a line passing through the point (3, 1) with slope 2. Does the point (4, 3) lie on this line? Solution Let m = 2 and (x1, y1) = (3,1) in the point-slope form. To determine whether the point (4, 3) lies on the line, substitute 4 for x and 3 for y. y – y1 = m(x – x1) y − 1 = 2(x – 3) 3 – 1 ? 2(4 – 3) 2 = 2 The point (4, 3) lies on the line because it satisfies the point-slope form.

Example Use the point-slope form to find an equation of the line passing through the points (−2, 3) and (2, 5). Solution Before we can apply the point-slope form, we must find the slope.

Example (cont) We can use either (−2, 3) or (2, 5) for (x1, y1) in the point-slope form. If we choose (−2, 3), the point-slope form becomes the following. If we choose (2, 5), the point-slope form with x1 = 2 and y1 = 5 becomes y – y1= m(x – x1)

Example Find an equation of the line passing through (4, 2) and (1, 5). Solution First find the slope of the line.

Example (cont) Find an equation of the line passing through (4, 2) and (1, 5). Solution Now substitute −1 for m and (4, 2) for x and y in the slope intercept form. The point (1, 5) could be used instead. The slope-intercept form is y = −x + 6.

Example Find a line y = mx + b that models the data in the table. Solution Carefully make a scatter plot of the data and sketch a line. x 1 2 3 4 5 f(x) 6 11 18 24 31

Example (cont) Find a line y = mx + b that models the data in the table. Solution Chose two points on the line and find the equation. The equation of the line is y = 6(x – 3) + 18 or y = 6x. x 1 2 3 4 5 f(x) 6 11 18 24 31

The equation of a horizontal line with y-intercept b is y = b. EQUATIONS OF HORIZONTAL AND VERTICAL LINES The equation of a horizontal line with y-intercept b is y = b. The equation of a vertical line with x-intercept h is x = h.

Example Find equations of the vertical and horizontal lines that pass through the point (−5, 2). Graph these two lines. Solution The x-coordinate of the point (−5, 2) is −5. The vertical line x = −5 passes through every point on the xy-plane including the point (−5, 2). Similarly, the horizontal line y = 2 passes through every point with a y-coordinate of 2 including the point (−5, 2).

Example (cont) Find equations of the vertical and horizontal lines that pass through the point (−5, 2). Graph these two lines. Solution x = −5 y = 2

PARALLEL LINES Two lines with the same slope are parallel. Two nonvertical parallel lines have the same slope.

Example Find the slope-intercept form of a line parallel to y = 3x + 1 and passing through the point (2, 1). Solution The line has a slope of 3 any parallel line also has slope 3.

PERPENDICULAR LINES Two lines with nonzero slopes m1 and m2 are perpendicular if m1m2 = −1. If two lines have slopes m1 and m2 such that m1 · m2 = −1, then they are perpendicular. A vertical and horizontal line are perpendicular.

Example Find the slope-intercept form of the line perpendicular to y = x – 3 passing through the point (4, 6). Solution The line has slope m1 = 1. The slope of the perpendicular line is m2 = −1. The slope-intercept form of a line having slope −1 and passing through (4, 6) can be found as follows.

Example Find the slope-intercept form of each line shown. Verify that the two lines are perpendicular. Solution The graph of y1 has the slope m1 = 3/2 because the line rises 3 units for every 2 units of run. It’s y-intercept is −1/2, and it’s slope intercept form is The graph of y2 has the slope m2 = −2/3 because the line falls 2 units for every 3 units of run. It’s y-intercept is 1/3, and it’s slope intercept form is y2 y1

Example (cont) Find the slope-intercept form of each line shown. Verify that the two lines are perpendicular. Solution The graph of y2 has the slope m2 = −2/3 because the line falls 2 units for every 3 units of run. It’s y-intercept is 1/3, and it’s slope intercept form is y2 y1