Equations of Lines in the Coordinate Plane Section 3.7 p.189

Graphing Linear Equations Definitions: Cartesian Coordinate Plane – a graph X – axis – the horizontal axis of a coordinate plane Y – axis – the vertical axis of a coordinate plane

Graphing Linear Equations Definitions: Origin – where the two axes meet (0,0) Ordered pair – x and y values of a point on a graph Also called a point of a set of coordinates Quadrants – the four sections that the x and y axes divide the coordinate plane into – named I, II, III, and IV

Coordinate Plane Identify Origin Y-axis X-axis Quadrants I, II, III, and IV

“Rise Up Run Out” Slope “Steepness” What are some examples where slope is a factor? grade of a road, incline of wheelchair ramp, pitch of a roof, etc.

Slope of a Line Slope m= 𝑟𝑖𝑠𝑒 𝑟𝑢𝑛 = 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑐ℎ𝑎𝑛𝑔𝑒 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑐ℎ𝑎𝑛𝑔𝑒 = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 = ( 𝑦 2− 𝑦 1 ) ( 𝑥 2 − 𝑥 1 ) Pick any two points on a line to compute the slope

Determine the slope of a line given the coordinates of two points on the line Given A (-1,2) and B (4, -2) Find the slope of line AB 𝑚=− 4 5

Find the slope of the segment below (5, 4) and (3, -1) Slope = m= =

Positive vs. negative slope Positive slope- rises to the right Negative slope- falls to the right

Slope Horizontal line ∆x ∆x Vertical line ∆x 0 Slope = ∆y = 0 = 0 Slope = ∆y = ∆y = undefined ∆x 0

Given C (4, 0) and D (4, -2) Find the slope of line CD 𝑚= 2 0 undefined

Slope of a Line Special cases: x = 4 What will this slope be? y = - 3

Slope-Intercept Form 𝑦=𝑚𝑥+𝑏 𝑚=𝑠𝑙𝑜𝑝𝑒 𝑎𝑛𝑑 𝑏=𝑦−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 Given 𝑦= 2 3 𝑥−5, What is the slope? 2 3 What are the coordinates of the y-intercept? (0, -5)

Graph y = 2 3 x -5

Point-Slope Form 𝑦− 𝑦 1 = 𝑚(𝑥− 𝑥 1 ) 𝑚=𝑠𝑙𝑜𝑝𝑒 𝑎𝑛𝑑 𝑥 1 , 𝑦 1 𝑖𝑠 𝑎 𝑝𝑜𝑖𝑛𝑡 𝑜𝑛 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒 Given point A (3, 5) on the line with a slope of -1, find the equation of the line in point-slope form. 𝑦−5= −1(𝑥−3) Write the equation of this line in slope-intercept form. 𝑦= −𝑥+8

What is the equation of a line in point-slope form passing through point A(-2,-1) and B(3, 5)? First find the slope; 𝑚= 6 5 Then plug one of the points into the point-slope form of the line; 𝑦−5= 6 5 𝑥−3

More Practice What is the equation of a line in slope intercept form with slope of -2 and a y-intercept of (0, 5)? 𝑦=−2𝑥+5 In point-slope form? y- 5 = -2(x-0) What is the equation in point-slope form of the line through (-1, 5) with a slope of 2? 𝑦−5=2(𝑥+1) In slope-intercept form? 𝑦=2𝑥+7

Homework P.194-195 #9-41 odd Additional Practice 13-2 Slope of a Line worksheet

How do you think the slopes of parallel lines compare? What about perpendicular lines?

Slopes of Parallel Lines (GSP)

Slopes of Perpendicular Lines (GSP)

3.8 Slopes of Parallel and Perpendicular Lines Two non-vertical lines are parallel if and only if their slopes are equal. (parallel lines have the same slope) Two non-vertical lines are perpendicular if and only if the product of their slopes is -1 (slopes of perpendicular lines are negative reciprocals of each other) m1 *m2 = -1 or m1= -1/m2

Are the two lines below parallel? y= -3x +4 and y=-3x -10 y= 4x-10 and y=2x-10 y= 2 3 x +5 and y = 2 3 x +7 Are the two lines below perpendicular? y= 4x – 2 and y= - 1 4 x +5 y= - 3 4 x +4 and y= 4 3 x +4 y= 3 4 x -10 and y= 4 3 +5

Slope = (-1 – 3)/ (5 – (-3)) = -4/8 = -1/2 A. slope = -1/2 Given a line through points (5,-1) and (-3, 3), find the slope of all lines A. parallel to this one B. perpendicular to this one Slope = (-1 – 3)/ (5 – (-3)) = -4/8 = -1/2 A. slope = -1/2 B. slope = 2

Are the two lines below perpendicular? (-4, 2) and (0, -4) (-5, -3) and (4, 3)

Homework p.201-203 #7-10, 15-18, 23, 25, 31, 33 13-3 Parallel and Perpendicular Lines worksheet 13-7 Writing Linear Equations worksheet #11-23 odd, 24-26 all

Find the distance between points A and B Two points in a horizontal line Distance = absolute value of the difference in the x-coordinates Distance=|-2 – 2| = 4 or |2 – (-2)| = 4

Find the distance between points A and B Two points in a vertical line Distance = absolute value of the difference in the y-coordinates Distance=|-8 – 3| = 11 or |3 – (-8)| = 11

What about two points that do not lie on a horizontal or vertical line? How can you find the distance between the points? The distance between two points is equal to the length of the segment with those points as the endpoints

The Distance Formula The distance between points (x1, y1) and (x2, y2) is given by: d = Find the distance between (0, 0) and (7, 24) d = 25

Midpoint Formula Review Find the midpoint of the line segment with endpoints (4, 7) and (-2, 5) 𝑥1+𝑥2 2 , 𝑦1+𝑦2 2 4+(−2) 2 , 7+5 2 (1, 6)

Class work 13-1 Distance Formula worksheet 13-5 Midpoint Formula worksheet