Math 025 Section 7.3 Slope

Objectives: to find the x- and y-intercepts of a straight line to graph a line using x- and y-intercepts to find the slope of a straight line to determine from their slopes whether two lines are parallel to graph a line using the slope and the y-intercept

The graph of 2x + 3y = 6 is shown below. The graph crosses the x-axis at the point (3, 0) y This point is called the x-intercept of the graph (0, 2) (3, 0) x The graph crosses the y-axis at the point (0, 2) This point is called the y-intercept of the graph Note: y = 0 at the x-intercept x = 0 at the y-intercept

Find the x-intercept and y-intercept for x – 2y = 4 Then graph the line x-intercept: Let y = 0 y x – 2(0) = 4 x = 4 x-intercept: (4, 0) y-intercept: Let x = 0 x 0 – 2y = 4 -2y = 4 y = -2 y-intercept: (0, -2)

The slope of a line is a measure of the slant of the line. It is sometimes described as: Slope = rise run It can also be described as: Slope = change in y change in x positive slope zero slope negative slope undefined slope

D y slope = m = D x If you move in a negative direction to find Dy or Dx, use a negative number to describe that movement. Red line: -3 m = = -3 1 Blue line: 5 7 5 m = 7

Find the slope of the line containing the given points You can find the slope of a line without graphing it if you know two points, P1(x1, y1) and P2(x2, y2), that are on the line. y2 – y1 slope formula: m = x2 – x1 Find the slope of the line containing the given points -3 – 3 -6 -3 P1(1, 3) P2(5, -3) m = = = 5 – 1 4 2 -3 – 2 -5 P1(-1, 2) P2(-1, -3) m = = = Undefined -1 – (-1)

The slopes are the same, so the lines are parallel. Parallel lines have the same slope Determine whether the line through P1 and P2 is parallel to the line through Q1 and Q2 -5 + 9 4 – 6 4 -2 P1(4, -5) P2(6, -9) mP = = = -2 -4 – 4 5 – 1 -8 Q1(5, -4) Q2(1, 4) mQ = = = -2 4 The slopes are the same, so the lines are parallel.

The slopes are negative reciprocals, so the lines are perpendicular. Perpendicular lines have slopes that are negative reciprocals of each other Determine whether the line through P1 and P2 is perpendicular to the line through Q1 and Q2 -5 + 9 -4 + 6 4 2 P1(-4, -5) P2(-6, -9) mP = = = 2 1 – 5 4 + 4 -4 -1 Q1(-4, 5) Q2(4, 1) mQ = = = 8 2 The slopes are negative reciprocals, so the lines are perpendicular.

Determine whether the lines through P1 and P2 and through Q1 and Q2 are parallel, perpendicular or neither 1 + 2 5 – 3 3 2 mP = = P1(5, 1) P2(3, -2) -4 + 2 3 – 0 -2 Q1(0, -2) Q2(3, -4) mQ = = 3 Perpendicular 1 -2 -1 + 2 1 – 3 mP = = P1(1, -1) P2(3, -2) -5 – 1 2 + 4 -6 Q1(-4, 1) Q2(2, -5) mQ = = = -1 6 Neither

The slope is the coefficient of x (-2, 5) y = -3x + 2 2 Notice that the above equation can be used to determine the slope and the y-intercept without having seen the graph. (2, -1) The slope is the coefficient of x 5 + 1 -2 – 2 6 -4 -3 m = = = 2 The y-coordinate of the y-intercept point is the constant at the end of the equation. y-intercept = (0, 2)

Slope = m y-intercept = (0,b) Slope-intercept form of an equation y = mx + b Slope = m y-intercept = (0,b) Find the slope and the y-intercept of each equation y = -3x + 5 4 2x – 5y = 10 -5y = -2x + 10 y = 2x - 2 5 slope = -3 4 slope = 2 5 y-intercept = (0,5) y-intercept = (0, -2)

Graph: y = -2x + 4 5 slope = -2/5 y-intercept = (0, 4) 1st: Graph the y-intercept point 2nd: Use the slope movements to find another point 3rd: Draw the line

Graph: 2x – 3y = -3 -3y = -2x – 3 y = 2x + 1 3 slope = 2/3 y-intercept = (0, 1)