Objective Students will find the slope of a line using 2 points.

Vocabulary Rise Slope = Run Slope describes the steepness of a line. Slope is the ratio of the rise (or vertical change) to the run (or the horizontal change) Vertical Change Rise Slope = Run Horizontal Change

4.4 Slope 4 2 2 Rise Slope = Run Slope (m) = = You get the x and y-values from any two points on the line. In this instance the ordered pairs are listed Rise Slope = x y (1, 2) (-1,-2) Run 4 Slope (m) = 2 = 2

4.4 Slope 2 1 2 Rise Slope = Run Slope (m) = = You get the x and y-values from any two points on the line. In this instance the ordered pairs are listed Rise Slope = x y (1, 1) (-1,-1) Run 2 Slope (m) = = 1 2

4.4 Slope 3 −𝟑 -1 Rise Slope = Run Slope (m) = = You get the x and y-values from any two points on the line. In this instance the ordered pairs are listed Rise x y Slope = Run 3 Slope (m) = −𝟑 = -1

Finding Slope with a Formula The formula for slope is: y2 – y1 x2 – x1 m =

Example 𝟒 −𝟓 y2 – y1 x2 – x1 (−𝟏,𝟑) (𝟑,−𝟐) (−𝟐) – 3 3 – (−𝟏) You get the x and y-values from any two points on the line. In this instance the ordered pairs are listed y2 – y1 x2 – x1 m = 1. Write the formula (−𝟏,𝟑) (𝟑,−𝟐) (−𝟐) – 3 3 – (−𝟏) m = (x1,y1) (x2,y2) 2. Substitute −𝟓 𝟒 m = 3. Simplify

1. Find the slope of a line passing through the points (−𝟑, −𝟐) 𝒂𝒏𝒅 (𝟏, 𝟔) 2. Find the average rate of change of a line passing through the points (𝟐, −𝟔) 𝒂𝒏𝒅 (𝟗, −𝟏) (𝒙𝟏 , 𝒚𝟏) (𝒙𝟐, 𝒚𝟐) (𝒙𝟏, 𝒚𝟏) ( 𝒙𝟐, 𝒚𝟐) y2 – y1 x2 – x1 m = y2 – y1 x2 – x1 m = 6 – (- 2) 1 – (- 3) m = −𝟏 – (−𝟔) 9 – 2 m = 𝟓 𝟕 8 4 m = 5 7 m = 2

1. Find the slope of a line passing through the points (−𝟏, −𝟑) 𝒂𝒏𝒅 (𝟑, 𝟏) 2. Find the average rate of change of a line passing through the points (𝟔, 𝟓) 𝒂𝒏𝒅 (−𝟖, 𝟓) (𝒙𝟏 , 𝒚𝟏) (𝒙𝟐, 𝒚𝟐) (𝒙𝟏, 𝒚𝟏) ( 𝒙𝟐, 𝒚𝟐) y2 – y1 x2 – x1 m = y2 – y1 x2 – x1 m = 1 – (−𝟑) 3 – (−𝟏) m = 𝟓 – 𝟓 −𝟖−𝟔 m = 𝟒 -14 m = 𝟎 𝟏 4 m =

You decide to go on a diet. On the first day you weighed 100 lbs You decide to go on a diet. On the first day you weighed 100 lbs., and on the fifth day you weighed 97 lbs. Find the average rate of change for your diet plan passing through the points (1, 100) and (5, 97) (𝒙𝟏, 𝒚𝟏) ( 𝒙𝟐, 𝒚𝟐) y2 – y1 x2 – x1 m = 1. Write the formula 97 – 100 5 – 1 m = 2. Substitute −𝟑 4 m = 3. Simplify − 𝟑 𝟒 𝒑𝒐𝒖𝒏𝒅𝒔 𝒂 𝒅𝒂𝒚

Visual Representation of Slope You can tell the slope of a line just by looking at it… The slope of a line can be either positive, negative, zero or undefined…

Positive Slope x y

Negative Slope x y

Zero Slope x y

Undefined Slope x y

You are saving money for retirement You are saving money for retirement. After the first year you have $3000, and after the tenth year you have $21,000. Find the average rate of change for your retirement savings. 1. 𝟏 , 𝟑𝟎𝟎𝟎 (𝟏𝟎, 𝟐𝟏𝟎𝟎𝟎) 21000 – 3000 10 – 1 m = (𝒙𝟏, 𝒚𝟏) ( 𝒙𝟐 , 𝒚𝟐) $𝟐𝟎𝟎𝟎 𝒂 𝒚𝒆𝒂𝒓 18000 9 m = 2. Find the slope of a line passing through the points ( 6, 1) and ( 6, 4) (𝒙𝟏, 𝒚𝟏) ( 𝒙𝟐, 𝒚𝟐) 𝑼𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅 3 m = 4 – 1 6 – 6 m =

You are saving money for college You are saving money for college. After the first year you have $3000, and after the thirteenth year you have $39,000. Find the average rate of change for your college savings. 1. 𝟏 , 𝟑𝟎𝟎𝟎 (𝟑, 𝟑𝟗𝟎𝟎𝟎) 39000 – 3000 13 – 1 m = (𝒙𝟏, 𝒚𝟏) ( 𝒙𝟐 , 𝒚𝟐) $𝟑𝟎𝟎𝟎 𝒂 𝒚𝒆𝒂𝒓 36000 12 m = 2. Find the slope of a line passing through the points ( 2, 1) and ( 4, −𝟒) (𝒙𝟏, 𝒚𝟏) ( 𝒙𝟐, 𝒚𝟐) − 𝟓 𝟐 −𝟓 2 m = – 𝟒 – 1 4 – 2 m =

4.4 Slope Find the value of y. ( 0, y) and ( 2, 5) m=2 (x1,y1) (x2,y2) y2 – y1 x2 – x1 m = 1. Write the formula 5 – y 2 – 0 2 = 2. Substitute 𝟒= 𝟓 − 𝒚 5 - y 2 = −𝟓 −𝟓 (2) (2) −𝟏=− 𝒚 𝟏= 𝒚

Find the value of y. ( 0, - 2) and ( 2, y) 𝒎=𝟑 1. 2. Find the value of y. (- 3, - 3) and ( -2, y) 𝒎=𝟓 (x1,y1) (x2,y2) (x1,y1) (x2,y2) y2 – y1 x2 – x1 m = y2 – y1 x2 – x1 m = y – (-3) -2 – (-3) 5 = y – (-2) 2 – 0 3 = y + 3 1 5 = y + 2 2 3 = (2) (2) 𝟓= 𝒚+𝟑 𝟔= 𝒚+𝟐 −𝟑 −𝟑 −𝟐 −𝟐 𝟐= 𝒚 𝟒= 𝒚

4.4 The Slope of a line 1. Slope describes the steepness of a line & is found with the ratio: Slope (m)= 𝑹𝒊𝒔𝒆 𝑹𝒖𝒏 = (𝒉𝒆𝒊𝒈𝒉𝒕 𝒃𝒆𝒕𝒘𝒆𝒆𝒏 𝒑𝒐𝒊𝒏𝒕𝒔) (𝒍𝒆𝒏𝒈𝒕𝒉 𝒃𝒆𝒕𝒘𝒆𝒆𝒏 𝒑𝒐𝒊𝒏𝒕𝒔) →→• 𝒂 • 𝒃 𝑹𝒊𝒔𝒆 𝑹𝒖𝒏 = 𝟐 𝒖𝒑 𝟐 𝒓𝒊𝒈𝒉𝒕 → +𝟐 +𝟐 =𝟏 2. There are four types of slopes: • Positive —Rises up to the right Negative —Drops down to the right Zero —Flat or horizontal Undefined—vertical (broken function)

4.4 The Slope of a line Positive Negative Zero Undefined (We’ve seen these before) 𝑼𝒑 𝑹𝒊𝒈𝒉𝒕 = + + 𝑼𝒑 𝑳𝒆𝒇𝒕 = + − 𝐲=# 𝒙=# (or) (or) 𝑫𝒐𝒘𝒏 𝑳𝒆𝒇𝒕 = − − 𝑫𝒐𝒘𝒏 𝑹𝒊𝒈𝒉𝒕 = − + 𝑹𝒊𝒔𝒆 𝟎 𝑹𝒖𝒏 # 𝑹𝒊𝒔𝒆 # 𝑹𝒖𝒏 𝟎 Both are Both are 0 Can’t ÷𝟎! 𝒎=+ 𝒎=− NO Slope x y x y x y x y

4.4 The Slope of a line 1. Slope can be found without a graph if you know 2 points from the line. Slope (m)= 𝑹𝒊𝒔𝒆 𝑹𝒖𝒏 = 𝑪𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒚 𝑪𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒙 = 𝒚 𝟐 − 𝒚 𝟏 𝒙 𝟐 − 𝒙 𝟏 *Find m given (3,2) & (-4,1) (𝒙 𝟏 , 𝒚 𝟏 ) (𝒙 𝟐 , 𝒚 𝟐 ) 𝒎= 𝟏−𝟐 −𝟒−𝟑 = −𝟏 −𝟕 = 𝟏 𝟕 𝒐𝒓 𝒎= 𝟐−𝟏 𝟑−(−𝟒) = 𝟏 𝟕