Lesson 2 4 Core Focus on Linear Equations

Lesson 2. 4 Core Focus on Linear Equations Rate of Change

Warm-Up 1. Write a recursive routine for a sequence of increasing numbers with a negative start value. Example: 2. Julia climbs 32 feet in elevation during each minute of her hike. She starts the hike at an elevation of 110 feet. Write a recursive routine that describes her elevation based on the number of minutes she has been hiking. SV = 110, Operation = Add 32

Lesson 2. 4 Rate of Change Calculate rates of change and start values.

Vocabulary Rate of Change The change in the output divided by the change in the input. Good to Know! So far in Block 2, you have been given the operation that allows you to move from one number in the sequence to the next. There are many situations where the operation is not given for just one step. For example: Colton eats 560 calories in 10 minutes. Luke paints 72 pictures in 3 days. Karen goes down 90 steps in 4 minutes. In order to determine the operation needed for each situation, you must calculate the rate of change.

Good to Know! The rate of change can be found by calculating the change in the output (or y-values) divided by the change in input (x-values). Rate of change is also called unit rate. It is important to think about which term is being used as the ‘counter’ and place that term in the denominator of the rate because it is your x-value. Time is the most common ‘counter. ’ It is important to determine if a situation is giving you increasing or decreasing numbers in the recursive sequence. This will help you decide if you are adding or subtracting your “rate of change” amount. Situation Rate of Change Operation Colton eats 560 calories in 10 minutes. Add 56 Karen goes down 90 steps in 4 minutes. Subtract 22. 5

Example 1 Determine the rate of change for the situation. State the operation that would occur in the recursive routine. a. Jessica loses $580 in 20 days in the stock market. The independent variable is the number of days. This term goes in the denominator. She is losing money, so the operation involves subtraction. Operation = Subtract $29

Example 1 Continued… Determine the rate of change for the situation. State the operation that would occur in the recursive routine. b. Patrick earned $53 for delivering 10 packages. The independent variable is the number of packages. He is earning money, so the operation involves addition. Operation = Add $5. 30

Extra Example 1 Determine the rate of change for each situation. a. A flower grew 15 inches in 5 weeks. +3 inches per week b. An elevator descended 40 feet in 4 seconds. − 10 feet per second

Rate of Change The rate of change is the change in y-values over the change in x-values. Good to Know! In some situations, information will be given to you in an input-output table. In those cases, you must be able to locate numbers on the table that will allow you to determine the rate of change. Once the rate of change is determined, locate or calculate the start value from the table. The start value is the y-value that is paired with the x-coordinate of zero.

Example 2 Determine the rate of change and start value from the input-output table. Choose two ordered pairs. Look for two consecutive numbers in the ‘counter’ column. Change in x-values = +1 Change in y-values = +3 Calculate the rate of change. The start value is 4 because it is the y-value that is paired with an x-value of 0.

Extra Example 2 Determine the rate of change and the start value for the table. start value = 17 rate of change = − 2

Example 3 The rate of change in the table is − 2. Find the start value. 10 8 Rewrite the table to include the x-coordinates to 0. The rate of change is – 2. Work backwards to get the x-coordinate of 0 by doing the opposite of the rate of change. Add 2 for each step. The start value is 10.

Extra Example 3 The rate of change in the table is + 5. Find the start value = 24

Example 4 Find the start value and the rate of change for the input-output table. Find the rate of change by selecting two pairs of numbers. Find the change in x and the change in y. Calculate the rate of change. Change in x-values = +2 Change in y-values = +8

Example 4 Continued… Find the start value and the rate of change for the input-output table. Use the rate of change to work forwards from x = – 2 to find the y-value paired with the x-coordinate of 0. That start value is – 1. The rate of change is +4.

Extra Example 4 Determine the rate of change and the start value for the table. start value = 10 rate of change = − 9

Communication Prompt How does rate of change relate to a recursive routine?

Exit Problems Determine the rate of change and start value for each table. 1. x 0 1 2 3 4 y − 5 1 7 13 19 ROC = +6 SV = − 5 2. x − 1 1 3 6 7 y 22 14 6 − 10 ROC = − 4 SV = 18