Holt CA Course 1 7-6Rate of Change and Slope SLOPE

Holt CA Course 1 7-6Rate of Change and Slope Slope Essential Questions How can slope be found? How are lines classified based on their slope? How can slope be interpreted as a rate of change?

Holt CA Course 1 7-6Rate of Change and Slope Vocabulary rate of change rise Slope Run Positive slope Undefined Slope Negative Slope Zero slope

Holt CA Course 1 7-6Rate of Change and Slope A rate of change is a ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable. The rates of change for a set of data may vary or they may be constant.

Holt CA Course 1 7-6Rate of Change and Slope Slope is a measure of Steepness.

Holt CA Course 1 7-6Rate of Change and Slope Types of Slope Positive Negative Zero Undefined or No Slope

Holt CA Course 1 7-6Rate of Change and Slope The formula may sometimes be written as m =∆y. ∆x What is ∆ ?

Holt CA Course 1 7-6Rate of Change and Slope The letter “m” is used to represent slope. Why?

Holt CA Course 1 7-6Rate of Change and Slope If given 2 points on a line, you may find the slope using the formula m = y 2 – y 1 x 2 – x 1

Holt CA Course 1 7-6Rate of Change and Slope The constant rate of change of a line is called the slope of the line.

Holt CA Course 1 7-6Rate of Change and Slope What if the numerator is 0? What if the denominator is 0?

Holt CA Course 1 7-6Rate of Change and Slope Divide both sides by 2. Find the value of a. A. Multiply. Check It Out! Example 3 slope = 2525 rise run y 0 x a 10 = 2525 a 2a = 5 ∙ 10 2a = 50 a = 25

Holt CA Course 1 7-6Rate of Change and Slope (5, 4) (1, 2) Then count horizontally to the second point to find the run. Find the slope of the line. Begin at one point and count vertically to find the rise. Additional Example 2: Finding the Slope of a Line slope = = The slope of the line is. 1212

Holt CA Course 1 7-6Rate of Change and Slope (3, 2) (–1, –2) Then count horizontally to the second point to find the run. Find the slope of the line. Begin at one point and count vertically to find the rise. Check It Out! Example 2 slope = = The slope of the line is 1.

Holt CA Course 1 7-6Rate of Change and Slope Divide both sides by 2. Find the value of a. A. Multiply. Additional Example 3: Finding a Rise or Run slope = 2323 rise run y 0 x a 6 = a6a 2a = 3 ∙ 6 2a = 18 a = 9

Holt CA Course 1 7-6Rate of Change and Slope Divide both sides by 2. Find the value of a. B. Multiply. Additional Example 3: Finding a Rise or Run slope = slope = rise run y 0 x a –4 – = 1212 a -4 2a = 1 ∙ 4 2a = 4 a = 2

Holt CA Course 1 7-6Rate of Change and Slope Divide both sides by 3. Find the value of a. B. Multiply. Check It Out! Example 3 slope = slope = rise run y 0 x a –6 = –1 3 a –6 3a = –1 ∙ –6 3a = 6 a = 2

Holt CA Course 1 7-6Rate of Change and Slope If you know any two points on a line, you can find the slope of the line without graphing. The slope of a line through the points (x 1, y 1 ) and (x 2, y 2 ) is as follows: y2 – y1y2 – y1x2 – x1x2 – x1y2 – y1y2 – y1x2 – x1x2 – x1 When finding slope using the ratio above, it does not matter which point you choose for (x 1, y 1 ) and which point you choose for (x 2, y 2 ). slope =

Holt CA Course 1 7-6Rate of Change and Slope Classification of lines by slope A line with positive slope: rises from left to right (m>0) A line with negative slope: falls from left to right (m<0) A line with slope of zero is horizontal (m=0) A line with undefined slope is vertical (m is undefined)

Holt CA Course 1 7-6Rate of Change and Slope Find the slope of the line that passes through B. (1, 3) and (2, 1). Additional Example 1: Finding Slope, Given Two Points Let (x 1, y 1 ) be (1, 3) and (x 2, y 2 ) be (2, 1). 1 – 3 2 – 1 Substitute 1 for y 2, 3 for y 1, 2 for x 2, and 1 for x 1. 22 1 = The slope of the line that passes through (1, 3) and (2, 1) is –2. = y 2 – y 1 x 2 – x 1 = –2 Simplify.

Holt CA Course 1 7-6Rate of Change and Slope Find the slope of the line that passes through A. (–2, –4) and (2, -2). Additional Example 1: Finding Slope, Given Two Points Let (x 1, y 1 ) be (–2, –4) and (x 2, y 2 ) be (2, -2). -2 – (-4) -2 – (–2) Substitute 6 for y 2, –3 for y 1, 4 for x 2, and –2 for x 1. The slope of the line that passes through (–2, –4) and (2, -2) is. = y 2 – y 1 x 2 – x 1 Simplify. =

Holt CA Course 1 7-6Rate of Change and Slope Find the slope of the line that passes through C. (3, –2) and (1, –2). Additional Example 1: Finding Slope, Given Two Points Let (x 1, y 1 ) be (3, –2) and (x 2, y 2 ) be (1, –2). –2 – (–2) 1 – 3 Substitute 2 for y 2, 2 for y 1, 1 for x 2, and 3 for x 1.  – 3 = The slope of the line that passes through (3, –2) and (1, –2) is 0. = y 2 – y 1 x 2 – x 1 = 0 Rewrite subtraction as addition of the opposite. 0 –2 =

Holt CA Course 1 7-6Rate of Change and Slope What is the slope of a vertical line? The line doesn’t run! All vertical lines have an undefined slope.

Holt CA Course 1 7-6Rate of Change and Slope Find the slope of the line that passes through A. (–4, –6) and (2, 3). Check It Out! Example 1 Let (x 1, y 1 ) be (–4, –6) and (x 2, y 2 ) be (2, 3). 3 – (–6) 2 – (–4) Substitute 3 for y 2, –6 for y 1, 2 for x 2, and –4 for x = The slope of the line that passes through (–4, –6) and (2, 3) is. 3 2 = y 2 – y 1 x 2 – x =

Holt CA Course 1 7-6Rate of Change and Slope Find the slope of the line that passes through B. (2, 4) and (3, 1). Check It Out! Example 1 Let (x 1, y 1 ) be (2, 4) and (x 2, y 2 ) be (3, 1). 1 – 4 3 – 2 Substitute 1 for y 2, 4 for y 1, 3 for x 2, and 2 for x 1. 33 1 = The slope of the line that passes through (2, 4) and (3, 1) is –3. = y 2 – y 1 x 2 – x 1 = –3 Simplify.

Holt CA Course 1 7-6Rate of Change and Slope Find the slope of the line that passes through C. (3, –2) and (1, –4). Check It Out! Example 1 Let (x 1, y 1 ) be (3, –2) and (x 2, y 2 ) be (1, –4). –4 – (–2) 1 – 3 Substitute 4 for y 2, 2 for y 1, 1 for x 2, and 3 for x 1. – – 3 = The slope of the line that passes through (3, –2) and (1, –4) is –2. = y 2 – y 1 x 2 – x 1 Simplify. = 1 –2 =

Holt CA Course 1 7-6Rate of Change and Slope Additional Example 2: Money Application The table shows the total cost of fruit per pound purchased at the grocery store. Use the data to make a graph. Find the slope of the line and explain what it shows. Graph the data. Pounds Cost Cost of Fruit

Holt CA Course 1 7-6Rate of Change and Slope You can use any two points to find the slope of the line. Helpful Hint

Holt CA Course 1 7-6Rate of Change and Slope Additional Example 2 Continued Find the slope of the line: The slope of the line is 3. This means that for every pound of fruit, you will pay another $3. y 2 – y 1 x 2 – x   5 = 3 Pounds Cost Cost of Fruit Substitute Multiply

Holt CA Course 1 7-6Rate of Change and Slope Check It Out! Example 2 The table shows the total cost of gas per gallon. Use the data to make a graph. Find the slope of the line and explain what it shows. Graph the data. Cost of Gas GallonsCost x y Gallons Cost of Gas Cost

Holt CA Course 1 7-6Rate of Change and Slope Check It Out! Example 2 Continued Find the slope of the line: The slope of the line is 2. This means that for every gallon of gas, you will pay another $2. = y 2 – y 1 x 2 – x  6 6  3 = x y Gallons Cost of Gas Cost Substitute. Multiply.

Holt CA Course 1 7-6Rate of Change and Slope The slope of a line may be positive, negative, zero, or undefined. You can tell which of these is the case by looking at the graphs of a line— you do not need to calculate the slope.

Holt CA Course 1 7-6Rate of Change and Slope

Holt CA Course 1 7-6Rate of Change and Slope