Summative Assessment for MAFS.8.F.1.2
- Compare the properties of the two linear functions described below and answer the questions.
Function 1: y = 4x + 8
Function 2:
x / y2 / 20
4 / 10
6 / 0
------Follow-up Questions ------
What is the y-intercept of each function?
- Function 1 has a y-intercept of 8
- Function 2 has a y-intercept of 30
How do you know? Answers will vary. Some students may express Function 2 in slope-intercept form (y = -5x +30) and use substitution, and some students may use a graph of the function and try to “eyeball” the y-intercept. Others may create input/output tables that work backwards to where x = 0.
Which function has a greater rate of change? How do you know?
- Function 2 has a greater rate of change because it has a “steeper” slope or greater absolute value than Function 1. This may be misleading to some students who think because 4 is positive compared to -5 that it must be greater, but the focus here in on the rate of change.
- Function 1 has a rate of change (slope) of 4. This can easily be found in slope-intercept form by looking at the 4 as the value that corresponds to the slope.
- Function 2 has a rate of change (slope) of -5. This can be found by determining that the ratio of the change in x on the input/output table = 2 and the change in y on the table is -10. Then when the unit rate is found for the ratio of these changes (-10/2) = -5
What does it mean if a rate of change is negative? What does it mean if it’s positive?
- When a rate of change is negative, as in these two functions, the values are decreasing overall. When the rate of change (slope) is positive, the values will be increasing overall.
- Compare the properties of the two linear functions described below and answer the questions.
Function 1: Regular Coffee Drinker
Stephanie has a gift card for coffee and she buys one regular cup of coffee each day. The table to the right shows some data about her gift card balance.
Function 2: Latte Drinker
Jack also has a gift card that started with a $50 balance. He buys a large latte each day for $3.95.
------Follow-up Questions ------
Which person started with a higher balance on their card? How do you know? What property of the functions tells you this starting amount?
- The Latte Drinker (Function 2) had a higher starting balance of $50, while the Regular Coffee Drinker (Function 1) only had a starting balance of $25.
- Answers will vary. This could be found by determining the rate of change (-1.25 per day) and then working backwards from the balance on day 1 (23.75 +1.25 = 25.00). For function 2, some students may write the function in slope intercept form (y = 3.95x +50) and then set x to zero. Or they might just use Common Sense because it is given.
- Both of these starting balances correspond to the y-intercept in the functions.
How long will each person be able to buy coffee using their gift card if their spending continues at the same rate?How do you know? What property of each function corresponds to this rate?
- The Regular Coffee Drinker (Function 1) will be able to buy coffee for 20 days, despite having a much lower starting balance. The Latte Drinker (Function 2) will only be able to buy lattes for 12 days before his balance gets too low.
- Answers will vary, but once students know the rate of change for each function and the starting balance, they can either set y=0 or they can reason through using a graph to “eyeball” it or a table of values.
- This coffee spending rate is also known as the rate of change or slope for each function.