Section 3.4 The Slope Intercept Form of a Linear Equation Chapter 3 Section 3.4 The Slope Intercept Form of a Linear Equation

Slope and the Slope - Intercept Form of a Line Similar Triangles have proportional sides, so no matter what two points on a line you pick: 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1 = 𝑦 3 − 𝑦 2 𝑥 3 − 𝑥 2 = 𝑦 3 − 𝑦 1 𝑥 3 − 𝑥 1 This number that remains constant for any line is called the Slope. It really measures how the line is "tilted". The letter m is often used to stand for the slope. 𝑠𝑙𝑜𝑝𝑒=𝑚= 𝑟𝑖𝑠𝑒 𝑟𝑢𝑛 = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥 = 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1 The line contains the points 0,2 and 1,5 . 𝑥 3 , 𝑦 3 𝑥 2 , 𝑦 2 Find Slope: 𝑚= 5−2 1−0 =3 𝑥 1 , 𝑦 1 1,5 0,2 The line also contains the points −1,−1 and 1,5 . Positive slope is an increasing function, Negative slope is a decreasing function. −1,−1 No matter what two points you pick on the line the slope is the same! What is the slope of the pictured line? Find Slope: 𝑚= 5− −1 1− −1 = 5+1 1+1 = 6 2 =3

Find the slope of the line pictured above. Find the slope of the line that goes through the points 2,4 and 5,6 . 𝑚= 6−4 5−2 = 2 3 Points: −6,2 and, 3,−2 So, 𝑚= −2−2 3− −6 = −2+ −2 3+6 = −4 9 =− 4 9 Find the value for d so that the line passing through the points −1,2 and 𝑑,8 will have a slope of 3 4 . 3 4 = 8−2 𝑑− −1 3 4 = 6 𝑑+1 3 𝑑+1 =6∙4 3𝑑+3=24 3𝑑=24−3 3𝑑=21 𝑑= 1 3 ∙21 𝑑=7 Find the slope of the line pictured above. Can also be done by counting: 𝑟𝑖𝑠𝑒 𝑟𝑢𝑛 = −4 9 =− 4 9

Lines with Positive, Negative, Zero, and Undefined Slopes Positive Slope Negative Slope Zero Slope Undefined Slope The line goes "up hill" as you go from left to right. The line goes "down hill" as you go from left to right. The line is horizontal. 𝑚= 𝑟𝑖𝑠𝑒=0 𝑟𝑢𝑛 The line is vertical. 𝑚= 𝑟𝑖𝑠𝑒 𝑟𝑢𝑛=0

Solve for y Solve for y slope is -4 fraction −4 1 slope is 3 2 Slope-Intercept Form of a Line The slope-intercept form of a line is the equation of a line written as 𝒚=𝒎𝒙+𝒃 where the number m (the coefficient of x) is the slope and the number b (the constant) is the y coordinate of the y-intercept. Graphing by locating the y-intercept then thinking of the slope as a fraction that represents rise over the run and count up/down and over. Solve for y Solve for y slope is -4 fraction −4 1 y-intercept is 13 slope is 3 2 y-intercept is 2

Rise is -2 and Run is 5 Slope is −2 5 y-intercept is -4 Equation is: 𝑦=− 2 5 𝑥−4 Rise is 7 and Run is 4 Slope is 7 4 y-intercept is 3 Equation is: 𝑦= 7 4 𝑥+3 This is a vertical line so that all of the x-coordinates are the same. Slope is undefined There is no y-intercept Equation is: 𝑥=−6