Lesson 5-1 Warm-Up

“Rate of Change and Slope” (5-1) What is “rate of change” How can you find the “rate of change”? rate of change: a unit rate that tells how fast the dependent variable (y) is changes in relation to the dependent variable (y) – In other words, “x changes _?_ much for every change of 1 in y” Since the independent variable is plotted on the x-axis (the x’s) and the dependent variable is plotted on the y-axis (the y’s), we can find the rate of change by using the following rule: Rule: Since rate of change is a linear function (forms a line when graphed), you can use two point on the line to find the vertical and horizontal changes, such as (x1, y1) and (x2,, y2). To find the vertical change, find the difference in the y terms (y2,- y1), and to find the horizontal change, find the difference in the x terms (x2,- x1) Example: The following graph shows the altitude of an airplane as it’s coming in for a landing. Find the rate of change.

“Rate of Change and Slope” (5-1) The rate of change is , which means the airplane descends feet every second.

Find the rate of change for each pair of consecutive mileage amounts. Rate of Change and Slope LESSON 5-1 Additional Examples For the data in the table, is the rate of change the same for each pair of consecutive mileage amounts? Fee for Miles Driven Miles Fee 100 $30 150 200 250 $42 $54 $66 Find the rate of change for each pair of consecutive mileage amounts.

change in cost Cost depends on the Rate of Change and Slope LESSON 5-1 Additional Examples (continued) change in cost Cost depends on the change in number of miles number of miles. rate of change = 42 – 30 12 54 – 42 12 66 – 54 12 150–100 50 200–150 50 250–200 50 = , The rate of change for each pair of consecutive mileage amounts is $12 per 50 miles. The rate of change is the same for all the data.

Rate of Change and Slope LESSON 5-1 Additional Examples Below is a graph of the distance traveled by a motorcycle from its starting point. Find the rate of change. Explain what this rate of change means.

vertical change change in distance horizontal change change in time Rate of Change and Slope LESSON 5-1 Additional Examples (continued) rate of change = vertical change change in distance horizontal change change in time 400 – 0 20 – 0 = Use two points. = Divide the vertical change by the horizontal change. 400 20 = Simplify. 20 The rate of change is 20 m/s. The motorcycle is traveling 20 meters each second.

“Rate of Change and Slope” (5-1) What is “slope” How can you find the “rate of change”? Slope: the rate of change of a line on a graph in a unit rate form Rule: You only need two point on the line to find sloe (just like you do for rate of change). You can either divide the difference of the y values by the difference of the x values or simply divide the rise (vertical distance between the two points) by the run (horizontal distance between the two points) as in the following example: Example: Find the slope of the line.

“Rate of Change and Slope” (5-1) How can you tell if a slope is positive, negative, 0 (no slope), or undefined (impossible)?

Find the slope of each line. Rate of Change and Slope LESSON 5-1 Additional Examples Find the slope of each line. a. slope = rise run 4 – 1 0 – 2 = = 3 –2 = – 3 2 The slope of the line is – . 3 2

rise b. slope = run = = = 2 The slope of the line is 2. (continued) Rate of Change and Slope LESSON 5-1 Additional Examples (continued) b. slope = rise run –1 – 1 –2 – (–1) = = –2 –1 = 2 The slope of the line is 2.

Find the slope of the line through E(3, –2) and F(–2, –1). Rate of Change and Slope LESSON 5-1 Additional Examples Find the slope of the line through E(3, –2) and F(–2, –1). slope = y2 – y1 x2 – x1 Substitute (–2, –1) for (x2, y2) and (3, –2) for (x1, y1). –1 – (–2) –2 – 3 = = – 1 5 = –5 Simplify. The slope of EF is – . 1 5

Find the slope of each line. Rate of Change and Slope LESSON 5-1 Additional Examples Find the slope of each line. a. slope = y2 – y1 x2 – x1 2 – 2 1 – (–4) = Substitute (1, 2) for (x2, y2) and (–4, 2) for (x1, y1). = 5 Simplify. = 0 The slope of the horizontal line is 0.

Substitute (2, 1) for (x2, y2) and (2, –4) for (x1, y1). = Rate of Change and Slope LESSON 5-1 Additional Examples (continued) b. slope = y2 – y1 x2 – x1 Substitute (2, 1) for (x2, y2) and (2, –4) for (x1, y1). = 1 – (–4) 2 – 2 = 5 Simplify. Division by zero is undefined. The slope of the vertical line is undefined.

1. Find the rate of change for the data in the table. Rate of Change and Slope LESSON 5-1 Lesson Quiz 1. Find the rate of change for the data in the table. $15 per ticket 2. Find the rate of change for the data in the graph. 3. Find the slope of the line. 400 calories per hour 4 3

4. Find the slope of the line through (3, –2) and (–2, 5). Rate of Change and Slope LESSON 5-1 Lesson Quiz 4. Find the slope of the line through (3, –2) and (–2, 5). 7 5 – 5. State whether the slope is zero or undefined. a. b. undefined