8th Grade · 8.F.A.2

Comparing functions in 8th grade

Comparing functions means looking at two rules and deciding which one grows faster, starts higher or gives a bigger output. In 8th grade the two functions are shown in different ways, for example one as a table and one as an equation. Your child learns to read the same facts from each form so the comparison is fair.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

8.F.A.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

What it means, in plain words

A function can be written as an equation, a table, a graph or a sentence in words. Comparing functions in 8th grade means taking two functions, each in a different form, and asking the same question about both.

Most questions are about comparing linear functions. In a linear function, the output changes by the same amount at every step, so its graph is a straight line. Two numbers describe it. The rate of change is how much the output goes up (or down) when the input goes up by 1. The initial value is the output when the input is 0, which is where the graph crosses the vertical axis.

The real skill is finding those two numbers in each form. In the equation $y = 3x + 5$ , the rate of change is 3 and the initial value is 5. In a table, the rate of change is the change in $y$ divided by the change in $x$ . On a graph, it is how far the line rises for each step of 1 to the right. Once your child has both numbers for both functions, the comparison takes one line.

Your child should be able to:

Find the rate of change of a function from an equation, a table, a graph or a description
Find the initial value from each of those forms
Decide which of two functions grows faster or starts higher
Compare the outputs of two functions at the same input

Example

Function A is y = 2x + 1. In Function B's table, x is 0, 1, 2 and 3, and y is 4, 7, 10 and 13. Which function has the greater rate of change, and which starts higher?

In Function A, the number multiplying $x$ is 2, so the rate of change is 2. The initial value is 1.
In Function B, $y$ goes up by 3 each time $x$ goes up by 1: $7 - 4 = 3$ , $10 - 7 = 3$ and $13 - 10 = 3$ . The rate of change is 3.
When $x$ is 0, $y$ is 4, so the initial value of Function B is 4.
Function B has the greater rate of change (3 against 2) and also starts higher (4 against 1).

Example

Candle A is 12 inches tall and burns down 2 inches each hour. Candle B's height in inches after x hours is y = 10 - 3x. Which candle burns down faster?

Candle A loses 2 inches each hour, so its rate of change is $-2$ inches per hour.
In $y = 10 - 3x$ , the number multiplying $x$ is $-3$ , so Candle B's rate of change is $-3$ inches per hour.
Candle B burns down faster, because it loses 3 inches each hour and Candle A loses 2.

Where children usually get stuck

In a table where $x$ goes 0, 2, 4, 6 and $y$ goes up by 6 in each row, your child says the rate of change is 6.
The child assumes $x$ always goes up by 1 from row to row. Here $x$ goes up by 2, so $y$ grows 6 for every 2, which is 3 for each 1.
What helps. Before reading any table, have your child check how much $x$ goes up from one row to the next. If it is not 1, divide the change in $y$ by the change in $x$ .
Your child gives the first output in a table as the initial value, even though the table starts at $x = 2$ .
The initial value was learned as "the first number". It is the output when $x$ is 0, and not every table starts at 0.
What helps. Ask, "What is $y$ when $x$ is 0?" If the table does not show it, work backward one row at a time using the rate of change.
Asked which function has the greater output at $x = 1$ , your child compares the two rates of change instead.
Rate of change, initial value and output are all numbers about the same function, and words like "greater" and "more" sound the same in every question.
What helps. Before any work, have your child underline what is being compared: how fast it grows, where it starts or the output at one input.

Try four practice questions

Real 8th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
The equation below is Function A.
$\displaystyle y = 2x + 5$
Function B starts at 3 and increases by 4 for each 1-unit increase in x.
Function A initial value
Function B initial value
Choose which function has the greater initial value: A, B, or Same.
Show the answer and how to solve it
Enter 5 for Function A, 3 for Function B, and A for the comparison because 5 is greater than 3.
1. 1Look at Function A. In $y = 2x + 5$ , the constant term is 5, so Function A's initial value is 5.
2. 2Function B says it starts at 3, so Function B's initial value is 3.
3. 3Compare the two initial values. Since 5 is greater than 3, choose A.
Common mistake. A common mistake is entering the rate of change instead of the initial value. Here, 2 and 4 tell how the functions change, but the initial values are 5 and 3.
The idea underneath. The initial value is the output when x = 0: in y = mx + b it is the constant term, and in words it is the starting amount.

2On level

Function A is shown in the table below. Function B is shown on the graph that follows. Carlos's worked comparison of the two rates of change is shown after the graph. Which statement best evaluates Carlos's reasoning?

Function A

x	y
0	1
2	5
4	9

Function B

Carlos's comparison

Step 1	$\displaystyle \text{Function A: from }(0,1)\text{ to }(2,5),\text{ }5-1=4,\text{ so the rate is }4$
Step 2	$\displaystyle \text{Function B: from }(0,-1)\text{ to }(3,8),\text{ }\frac{8-(-1)}{3-0}=\frac{9}{3}=3$
Conclusion	$\displaystyle \text{Since }4>3,\text{ Function A has the greater rate.}$

The student used the change in y for Function A but ignored the change in x. Function A's rate is 2, so Function B has the greater rate.

The student miscounted the rise or run on Function B's graph.

The student's comparison is correct.

The student compared initial values instead of rates.

Show the answer and how to solve it

The correct statement is that the student used the change in y for Function A but ignored the change in x. Function A's rate is 2 and Function B's rate is 3, so Function B has the greater rate of change.

1Start with Function A in the table. Use two rows, such as (0, 1) and (2, 5), to find its rate of change.
2Rate of change is change in y divided by change in x. Function A's rate is 2, not 4.
3Now look at Function B on the graph. The two marked points are (0, -1) and (3, 8).
4Find Function B's slope from those marked points. Note that 8-(-1)=9 because subtracting a negative adds.
5Carlos's graph work is correct, but his first line for Function A is not. He used only the change in y (5-1=4) and forgot to divide by the change in x (2-0=2). Function A's true rate is 2, and Function B's rate is 3, so Function B has the greater rate of change.

Common mistake. A common mistake is treating the change in y as the full rate of change. In a table, you must divide by the change in x too. Here, 5-1=4 over 2-0=2 gives a rate of 2, not 4.

The idea underneath. Rate of change is how much y changes for each 1 unit of x, so you must use both changes in a table and on a graph.

3On level
Function A is shown by the graph below.
Function A
Function B is shown by the table below.
x y
1 1
2 -1
3 -3
4 -5
At x = 1, which function has the greater output: A, B, or are they the same?
A
Same
B
A
C
B
Show the answer and how to solve it
Function A has the greater output: Function A gives 4 and Function B gives 1 at x = 1.
1. 1On Function A's graph, find x = 1. The line passes through the exact grid point (1, 4), so Function A's output is 4.
2. 2In Function B's table, find the row where x = 1. That row shows y = 1, so Function B's output is 1.
3. 3Compare the two outputs. Since 4 > 1, Function A has the greater output.
Common mistake. A common mistake is reading the line at x = 4 (y = 1) or x = 5 (y = 0) instead of at x = 1. Start at x = 1 on the horizontal axis, move up to the line, and then read the y-value.
The idea underneath. Both representations give an output for the same input, so the comparison works only when you match the same x-value in both places and then compare those two y-values.
4Stretch
Function A is shown on the graph. Use the graph and Function B's table to record the initial value of each function, then choose which function has the greater initial value.
Function A
x y
-2 5
-1 4
0 3
1 2
2 1
Initial value of A
Initial value of B
Greater initial value
Show the answer and how to solve it
Enter −2 for the initial value of A, 3 for the initial value of B, and choose B because 3 is greater than −2.
1. 1For Function A, the initial value is the y-value when x = 0. On the graph, that is where the line crosses the y-axis. The line crosses at −2, so the initial value of A is −2.
2. 2For Function B, find the row where x = 0 in the table. That row shows y = 3, so the initial value of B is 3.
3. 3Compare the two initial values: −2 and 3. Since 3 is greater than −2, Function B has the greater initial value.
Common mistake. A common mistake is reading a positive x-intercept from the graph as the initial value of A, or picking the first y-value in Function B's table instead of the row where x = 0.
The idea underneath. An initial value is the output when the input is 0, so on a graph it is the y-intercept and in a table it is the row where x = 0.

x	y
-2	5
-1	4
0	3
1	2
2	1
Initial value of A
Initial value of B
Greater initial value