Date: Topic: Lines and Slope (1.2) Definition of Slope The slope of the line through the distinct points (x1, y1) and (x2, y2) is where x2 – x1 = 0. y1 y2 y Change in y Change in x = Rise Run y2 – y1 x2 – x1 (x1, y1) x1 x2 (x2, y2) x Find the slope of the line with points (3, 2) and (-5, 5)

The Possibilities for a Line’s Slope Positive Slope x y m > 0 Line rises from left to right. Zero Slope x y m = 0 Line is horizontal. Negative Slope x y m < 0 Line falls from left to right. m is undefined Undefined Slope x y Line is vertical. m = 2

Slope-Intercept Form of the Equation of a nonvertical line with slope m and y-intercept b is y = mx + b Find the slope and the y-intercept of the line whose equation is 2x – 3y + 6 = 0. 2x – 3y + 6 = 0 + 3y +3y To isolate the y-term, add 3y on both sides. 2x + 6 = 3y 3y = 2x + 6 Reverse the two sides. (This step is optional.) 3 3 Divide both sides by 3. y = 2/3x + 2 The slope is 2/3. The y-intercept is 2.

Graphing y=mx+b Using the Slope and Y-Intercept Plot the y-intercept, b. Plot a second point using the slope, m, rise over run. Draw a line through the two points. Graph the line whose equation is: -5 -4 -3 -2 -1 1 2 3 4 5 y = 2/3 x + 2 The slope is 2/3. The y-intercept is 2. We need two points in order to graph the line: We can use the y-intercept, 2, to obtain the first point (0, 2). We plot the second point on the line by starting at (0, 2), the first point. Then move 2 units up (the rise) and 3 units to the right (the run). This gives us a second point at (3, 4).

Point-Slope Form of the equation of a nonvertical line of slope m that passes through the point (x1, y1) is y – y1 = m(x – x1) If given two points find the slope using the points, and use one of the coordinates in the equation Write the point-slope form of the equation of the line passing through (-1,3) with a slope of 4. Then solve the equation for y. Solution We use the point-slope equation of a line with m = 4, x1= -1, and y1 = 3. This is the point-slope form of the equation. y – y1 = m(x – x1) y – 3 = 4[x – (-1)] Substitute the given values. y – 3 = 4(x + 1) We now have the point-slope form of the equation for the given line. y – 3 = 4x + 4 We can solve the equation for y by applying the distributive property. +3 + 3 Add 3 to both sides. y = 4x + 7

Equations of Lines Point-slope form: y – y1 = m(x – x1) Slope-intercept form: y = mx + b Horizontal line: y = b Vertical line: x = a General form: Ax + By + C = 0

Complete Student Checkpoint Indicate whether each line has a positive, negative, zero, or undefined slope. Write the equation of the line in slope-intercept form. (Assume scale is 1) x-axis y-axis x-axis y-axis a b c d y-intercept 3 zero -3 undefined positive slope -1/2 6 negative equation:

Use the given conditions to write an equation in point-slope form and slope-intercept form. Passing through (-3,2) and (3,6). Send you answer to me using the calculator

Slope and Parallel Lines If two nonvertical lines are parallel, then they have the same slope. DAY 2 Write an equation of the line passing through (-3, 2) and parallel to the line whose equation is y = 2x + 1. Express the equation in point-slope form and y-intercept form. y = 2x + 1 -5 -4 -3 -2 -1 1 2 3 4 5 (-3, 2) Rise = 2 Run = 1 y – y1 = m(x – x1) y1 = 2 m = 2 x1 = -3 Parallel lines have the same slope. Because the slope of the given line is 2, m = 2 for the new equation. y – 2 = 2[x – (-3)] y – 2 = 2(x + 3) y – 2 = 2x + 6 Apply the distributive property. y = 2x + 8 This is the slope-intercept form of the equation.

Slope and Perpendicular Lines 90° Two lines that intersect at a right angle (90°) are said to be perpendicular. There is a relationship between the slopes of perpendicular lines. Slope and Perpendicular Lines If two nonvertical lines are perpendicular, then the product of their slopes is –1. (2/3) • (-3/2) = -1 Slopes are negative reciprocals of each other Find the slope of any line that is perpendicular to the line whose equation is 2x + 4y – 4 = 0. 4y = -2x + 4 y = -1/2x + 1 Slope is –1/2. Any line perpendicular to this line has a slope that is the negative reciprocal, 2.

You Try Write an equation of the line passing through (-3,6) and perpendicular to the line whose equation is y=1/3 x +4 Express in point-slope form and slope-intercept form. Write an equation of the line passing through (-2,5) and parallel to the line whose equation is y=3x+1. Express in point-slope form and slope-intercept form. perpendicular slope:

Graphs and Viewing Windows Bob purchased a house 8 years ago for $42,000. This year is was appraised at $67,500. A linear equation V=mt + b, 0 ≤ t ≤ 15, represents the value V of the house for 15 years after it was purchased. Determine m and b. Graph the equation and trace to estimate in how many years after the purchase of this house it will be worth $72,500 Write and solve an equation algebraically to determine how many years after purchase this house will be worth $74,000.