Warm Up Find the value of m. 1. 2. 3. 4. undefined

Slopes of Lines / Equations of Lines Sections 3-3 & 3

Slope of a Line A number that describes the steepness of the line. Any two points on a line can be used to determine the slope.

Example: Finding the Slope of a Line Use the slope formula to determine the slope of each line. AB AC AD CD

Example: Use the slope formula to determine the slope of JK through J(3, 1) and K(2, –1).

One interpretation of slope is a rate of change One interpretation of slope is a rate of change. If y represents miles traveled and x represents time in hours, the slope gives the rate of change in miles per hour.

Example: Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. WX and YZ for W(3, 1), X(3, –2), Y(–2, 3), and Z(4, 3) Vertical and horizontal lines are perpendicular.

Example: Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. KL and MN for K(–4, 4), L(–2, –3), M(3, 1), and N(–5, –1) The slopes are not the same, so the lines are not parallel. The product of the slopes is not –1, so the lines are not perpendicular.

Example: Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. BC and DE for B(1, 1), C(3, 5), D(–2, –6), and E(3, 4) The lines have the same slope, so they are parallel.

Lesson Quiz 1. Use the slope formula to determine the slope of the line that passes through M(3, 7) and N(–3, 1). m = 1 Graph each pair of lines. Use slopes to determine whether they are parallel, perpendicular, or neither. 2. AB and XY for A(–2, 5), B(–3, 1), X(0, –2), and Y(1, 2) 4, 4; parallel 3. MN and ST for M(0, –2), N(4, –4), S(4, 1), and T(1, –5)

Warm Up! – Graph the line that satisfies each condition 1) Slope = -4, passes 2) Contains A(-1, -3), parallel 3) Contains M(4, 1), passes through to line CD with C(-1, 6) and to line GH with G(0, 3) P (-2, 1) D(4, 1) and H(-3, 0) Slope of line CD is -1, so Slope of line GH is -1, so slope of || line is also -1. slope of line is 1.

Writing Equations of Lines Remember from Algebra that an equation of a line can be written given any of the following: The slope and y-intercept (slope-intercept form) The slope and the coordinates of a point on the line (point-slope form) or, The coordinates of two points on the line

Example 1 – Writing an equation of a line given slope and the y-intercept “Write an equation in slope-intercept form of the line with slope of -3 and y-intercept of 2.” y = mx + b y = -3x + 2

Example 2 – Writing an equation of a line given slope and a point on the line. “Write an equation in point-slope form of the line with slope of ½ that contains (1, -4).” y – y1 = m(x – x1) y + 4 = ½ (x – 1)

Example 3 – Writing an equation of a line given two points on the line. “Write an equation of the line that passes through (-1, 0) and (1, 2).” y – y1 = m(x – x1) y – 0 = 1(x - -1) y = x + 1

Try these examples on your own 1.) Write the equation of the line in slope-intercept form that has a slope of ½ and y-intercept of -4. 2.) Write the equation of the line in point-slope form that has a slope of -2 and passes through point (3, -4). 3.) Write the equation of the line AB in slope-intercept form where A(-4, 0) and B(2, -3).

Example 4 – Writing Equations of Parallel Lines “Write the equation of the line that passes through (1, 3) and is parallel to the line y = -2x + 3” y – y1 = m(x – x1) y = -2x + 5 Answer left in slope-intercept form

Example 5 – Writing Equations of Perpendicular Lines “Write the equation of the line that passes through (-4, 2) and is perpendicular to the line y = ¼x + 3” y – y1 = m(x – x1) y = -4x - 14 Answer left in slope-intercept form

Try these examples on your own 1.) Write an equation of the line that passes through K(5, 0) and is parallel to line AB where A(0, 2) and B(-4, 3). 2.) Write an equation of the line that passes through Z(-1, -3) and is perpendicular to line CD where C(1, 2) and D(4, 0).