Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
The same kind of relationship can arrive in four different outfits: an equation, a graph, a table or a sentence. 8.F.A.2 asks students to compare two functions when each is dressed differently. One cell plan might be written as c = 0.10t + 15, while a second is shown as a table of minutes and costs. Which one charges more per minute? Which has the higher starting fee? To answer, students must extract the same properties from each form.
For linear functions the key properties are the rate of change and the initial value. From an equation in y = mx + b form they are read directly. From a table, the rate is the change in output divided by the change in input, and the initial value is the output when the input is zero (which may need working backward). From a graph, they come from the steepness and the vertical intercept, and from a description, from the words 'per' and 'starting at'. Once every function is translated into these terms, the comparison is straightforward.
A table with larger numbers is not necessarily growing faster. Students must compute how much the output changes per unit of input.
Dividing the output 23 by the input 4 gives the rate only if the function is proportional. With a starting value, use the change between two rows.
If inputs jump from 2 to 5, the change in output must be divided by 3, not by 1.
The first row might be x = 1, not x = 0. The initial value is the output at x = 0, which may need extending the pattern backward.
Function A is y = 3x + 2. Function B is given by the table x: 1, 3, 5 and y: 7, 15, 23. Which function has the greater rate of change, and which has the greater initial value?
Answer: Function B has both the greater rate of change (4) and the greater initial value (3).
A 'which would you choose?' task works well: two job offers, one described in words and one as a graph, or two taxi fares, one as an equation and one as a table. Students must justify a choice, which forces them to compare the rate and starting value rather than eyeballing numbers.
Assessment items usually pair two different representations and ask about rate of change. Teach a consistent first move: rewrite each function as 'starts at ___ and changes by ___ per ___'. That sentence frame also prepares students for interpreting models in 8.F.B.4.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: B) Q
P has rate 5. Q's rate is (14 - 2) ÷ (2 - 0) = 6. Q has the greater rate of change.
Answer: 3
y increases by 6 when x increases by 2, so the rate is 6 ÷ 2 = 3.
Answer: 5
Going back from x = 2 to x = 0 subtracts 2 × 3 = 6, so y = 11 - 6 = 5.
Answer: A) Gym B has a lower monthly rate and a higher joining fee
Gym B's rate is (90 - 50) ÷ 4 = 10 dollars per month, lower than Gym A's 15. Its joining fee is 50, higher than Gym A's 30.
Answer: 1
C decreases by 2 per hour and D by 3 per hour, so D decreases 1 unit per hour faster.
A full lesson with slides, activities and an exit ticket on comparing functions in different forms, pitched to grade 8 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 8.F.A.2 with an answer key, ready in about a minute.
Make a worksheet →Turn comparing functions in different forms into a quiz students answer online that marks itself, with a class summary for you.
Build a test →The examples are mostly linear, because rate of change is constant and easy to compare. Comparing nonlinear functions in detail comes in high school.
Mainly the rate of change and the initial value, plus whether the function increases or decreases and, in context, which is greater for a given input.