3.11Comparing Functions
Functions can be presented as equations, tables, graphs, or verbal descriptions. This lesson gives you a systematic method to extract rate of change and initial value from any representation — so you can compare any two functions with confidence.
Why This Matters
Comparing functions is one of the most common SAT and ACT question types. You will be given two functions in different forms — a table and an equation, a graph and a verbal description — and asked which grows faster or starts higher. This lesson gives you a systematic method for every case.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
How Do You Compare Two Functions?
Find the rate of change and initial value of each function, even when the functions are shown in different forms. Compare the rates to determine which changes faster, and compare the initial values to determine which starts higher or lower.
Lesson Overview
To compare two functions, you need to extract the same information from each — regardless of how they are presented. The two key quantities are the rate of change (slope) and the initial value (y-intercept). Once you have both for each function, comparison is straightforward.
Key Vocabulary
Rate of Change
How much the output changes per unit increase in input. For a linear function, this is the slope m.
Example: f(x) = 3x + 1 has rate of change 3.
Initial Value
The output when the input is 0. For f(x) = mx + b, the initial value is b.
Example: f(x) = 3x + 1 has initial value 1.
Linear Function
A function with a constant rate of change. Its graph is a straight line.
Example: f(x) = 2x − 4
Nonlinear Function
A function whose rate of change is not constant. Its graph is not a straight line.
Example: f(x) = x² has rate of change that increases.
How to Compare Functions
3-Step Process
- Find the rate of change of each function. Use slope formula, Δy/Δx from a table, rise/run from a graph, or key words from a verbal description.
- Find the initial value of each function. Look for b in the equation, y at x = 0 in a table, the y-intercept on a graph, or the starting value in a verbal description.
- Compare and interpret. Which rate is greater? Which initial value is greater? At what x-value do they intersect?
Comparison Organizer
| Representation | Rate of Change | Initial Value | Increasing / Decreasing | Linear / Nonlinear | Notes |
|---|---|---|---|---|---|
| Equation f(x) = 2x + 3 | m = 2 | b = 3 | Increasing | Linear | Read directly |
| Table: x=0,1,2,3; y=1,4,7,10 | Δy/Δx = 3 | y at x=0 = 1 | Increasing | Linear | Check Δy constant |
| Graph: line through (0,−2), slope 1/2 | rise/run = 1/2 | y-int = −2 | Increasing | Linear | Read from graph |
| Verbal: "starts at 5, grows by 4 each step" | 4 | 5 | Increasing | Linear | Identify key words |
| Table: x=0,1,2,3; y=1,2,4,8 | Not constant | 1 | Increasing | Nonlinear | Δy not constant |
Comparing Linear vs. Nonlinear
How to tell if a table is linear:
- Check if Δy/Δx is constant for every consecutive pair of rows.
- If yes → linear. The function can be written as f(x) = mx + b.
- If no → nonlinear. The function cannot be written as f(x) = mx + b.
⚠ Check ALL consecutive pairs — not just the first two.
Linear: f(x) = 2x + 1
Constant Δy = 2 per step → straight line
Nonlinear: y = 1, 2, 4, 8, 16
Δy = 1, 2, 4 (not constant) → curved
Math Tips
Always extract slope and y-intercept from BOTH functions before comparing — don't compare just one quantity.
A function with a larger slope grows faster, but a function with a larger y-intercept starts higher. These are different comparisons.
To check if a table is linear, verify that Δy/Δx is constant for ALL consecutive pairs — not just the first two.
Worked Examples
Function A: f(x)=4x+1 vs Function B: g(x)=2x+5
Function A: f(x) = 4x + 1. Function B: g(x) = 2x + 5. Which has a greater rate of change? Which has a greater initial value?
Function A: slope = 4, y-intercept = 1.
Function B: slope = 2, y-intercept = 5.
Rate of change: A (4) > B (2). Function A grows faster.
Initial value: B (5) > A (1). Function B starts higher.
Function A is given by the table: x = 0, 1, 2, 3; y = 3, 7, 11, 15. Function B: g(x) = 5x − 1. Which function has a greater rate of change?
Function A from table: Δy/Δx = (7 − 3)/1 = 4. Rate of change = 4.
Function B: slope = 5.
Compare: 5 > 4. Function B has a greater rate of change.
Function A passes through (0, 6) and (3, 0). Function B is given by the table: x = 0, 1, 2, 3; y = 0, 2, 4, 6. Compare the rates of change and initial values.
Function A: m = (0 − 6) / (3 − 0) = −2. Initial value = 6.
Function B: Δy/Δx = 2/1 = 2. Initial value = 0.
Rate of change: A = −2, B = 2. Function B increases; Function A decreases.
Initial value: A (6) > B (0).
Function A: 'A savings account starts with $200 and grows by $50 per month.' Function B: g(x) = 60x + 100. After 10 months, which function has a greater value?
Function A: f(x) = 50x + 200.
Function B: g(x) = 60x + 100.
At x = 10: f(10) = 50(10) + 200 = 700. g(10) = 60(10) + 100 = 700.
They are equal at x = 10!
Determine whether the table x = 0, 1, 2, 3; y = 1, 3, 9, 27 represents a linear function.
Check Δy: 3 − 1 = 2, 9 − 3 = 6, 27 − 9 = 18.
The differences are not constant (2, 6, 18).
This is NOT a linear function — it is exponential.
Function A: f(x) = −3x + 12. Function B: g(x) = 3x − 4. At what x-value do they have the same output? Which function is greater for x > that value?
Set equal: −3x + 12 = 3x − 4 → 16 = 6x → x = 8/3 ≈ 2.67.
For x > 8/3: g has positive slope (increasing), f has negative slope (decreasing). g(x) > f(x).
Check: at x = 4: f(4) = 0, g(4) = 8. g > f. ✓
Common Mistakes
Comparing visible y-values without comparing rates of change.
A function that starts lower can eventually surpass one that starts higher if it has a greater slope.
Assuming every table is linear without checking.
Always verify that Δy/Δx is constant for all consecutive pairs before treating a table as linear.
Ignoring units when comparing real-world functions.
Units matter. A slope of 50 dollars/month is not the same as 50 cents/month.
Misreading graph scales and computing the wrong slope.
Read the scale carefully. Count grid squares, not just visual distance.
Comparing y-intercepts when the question asks about rate of change.
Rate of change = slope. Initial value = y-intercept. These are different quantities.
Assuming the function with the larger starting value always stays larger.
If the other function has a greater slope, it will eventually overtake the one with the larger starting value.
Guided Practice
Guided Practice Video: Comparing Functions
Watch the guided practice walkthrough for comparing functions, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Answers are in the Answer Key section.
Function A: f(x) = 5x + 2. Function B: g(x) = 3x + 8. Which has a greater rate of change? Which has a greater initial value? At what x-value are they equal?
Hint: Read slope and y-intercept directly from each equation. Set them equal to find the intersection.
Function A: table x = 0, 1, 2, 3; y = 4, 9, 14, 19. Function B: g(x) = 6x + 1. Which function has a greater rate of change?
Hint: Find Δy/Δx for Function A. Then compare to the slope of Function B.
A table shows: x = 0, 1, 2, 3; y = 2, 4, 8, 16. Is this a linear function? Explain.
Hint: Check whether Δy/Δx is constant for all consecutive pairs.
Interactive Practice — 5 Questions
f(x) = 6x + 2 and g(x) = 4x + 8. Which has a greater rate of change?
A table: x = 0, 1, 2, 3; y = 1, 4, 9, 16. Is this linear?
Function A: table x = 0, 1, 2, 3; y = 5, 9, 13, 17. Function B: g(x) = 5x + 2. Which has a greater rate of change?
f(x) = −3x + 15 and g(x) = 3x + 3. At what x-value are they equal?
Two functions both have slope 4. Function A has y-intercept 2; Function B has y-intercept 7. Describe how their graphs compare.
Independent Practice
Answers are in the Answer Key section.
Independent Practice
f(x) = 6x + 1 and g(x) = 4x + 9. Which has a greater rate of change? Which has a greater initial value?
f(x) = −2x + 10 and g(x) = 2x − 4. Which is increasing? Which is decreasing? At what x-value are they equal?
Table A: x = 0, 1, 2, 3; y = 2, 6, 10, 14. Table B: x = 0, 1, 2, 3; y = 5, 8, 11, 14. Which table has a greater rate of change?
Function A: table x = 0, 1, 2, 3; y = 0, 3, 6, 9. Function B: g(x) = 4x − 2. Which has a greater rate of change?
Function A: "starts at 10, decreases by 3 each step." Function B: f(x) = −2x + 8. Which decreases faster?
Determine whether x = 0, 1, 2, 3; y = 0, 1, 4, 9 is linear. Explain.
f(x) = 2x + 5 and g(x) = 2x − 3. Compare the rates of change and initial values. Do the lines ever intersect?
Function A passes through (0, 4) and (5, 14). Function B: g(x) = 3x − 1. Which has a greater rate of change?
A table: x = 0, 2, 4, 6; y = 3, 7, 11, 15. A second function: g(x) = 3x + 1. Compare rates of change and initial values.
f(x) = 10x + 2 and g(x) = 8x + 20. At what x-value does f(x) exceed g(x)?
★ Function A: table x = 0, 1, 2, 3; y = 1, 3, 9, 27. Function B: g(x) = 8x + 1. Which has a greater value at x = 3? Which grows faster for large x?
Challenge★ Two functions both pass through (0, 6). Function A has slope 3; Function B has slope −2. Write both equations. At what x-value is Function A's output double Function B's output?
Challenge