6.6Comparing Linear and Exponential Functions
Identify, compare, and choose between linear and exponential models using tables, graphs, equations, and real-world contexts.
Why This Matters
The SAT and ACT frequently ask you to compare linear and exponential models — from tables, graphs, equations, and real-world contexts. Knowing how to identify the type of change (additive vs. multiplicative) and predict long-term behavior is one of the highest-value skills in Algebra 1.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How Can You Tell Whether a Function Is Linear or Exponential?
Quick Answer
A linear function changes by a constant difference over equal input intervals. An exponential function changes by a constant ratio or percent over equal input intervals. Linear graphs form straight lines, while exponential graphs curve and may change more rapidly over time. Check the table, equation, graph, and context before choosing a model.
Lesson Overview
You have studied linear functions (Unit 3) and exponential functions (Lessons 6.4 and 6.5). This lesson brings them together. The central question is: given a table, graph, equation, or real-world description, how do you determine which type of function you are looking at — and which model is more appropriate? The key is understanding the difference between additive change (linear) and multiplicative change (exponential).
Learning Goals
- 1Identify linear and exponential functions from tables using first differences and common ratios.
- 2Identify linear and exponential functions from equations and graphs.
- 3Recognize functions that are neither linear nor exponential.
- 4Compare initial values and rates of change for linear and exponential functions.
- 5Describe short-term and long-term behavior of each function type.
- 6Choose and justify an appropriate model for a real-world situation.
- 7Explain model choices using mathematical evidence.
Key Vocabulary
Constant Difference
The same value is added (or subtracted) between consecutive outputs over equal input intervals. Signals a linear function.
Constant Ratio
Each output is multiplied by the same factor over equal input intervals. Signals an exponential function.
First Differences
The differences between consecutive y-values in a table: Δy = y₂ − y₁, y₃ − y₂, etc.
Common Ratio
The quotient of consecutive y-values in a table: r = y₂/y₁, y₃/y₂, etc.
Additive Change
Each output increases or decreases by a constant amount. Characteristic of linear functions.
Multiplicative Change
Each output is multiplied by a constant factor. Characteristic of exponential functions.
Initial Value
The output when x = 0; the y-intercept. In y = mx + b it is b; in y = ab^x it is a.
Crossover Point
The x-value where two functions have equal outputs. Can be estimated from a table or graph.
Long-Term Behavior
How a function behaves as x grows very large. Exponential growth eventually dominates linear growth.
Model Selection
Choosing between linear and exponential (or another) function based on the type of change in the data.
Key Concepts
Linear Functions — Additive Change
A linear function has the form y = mx + b. The variable x has exponent 1 and is multiplied by a constant slope m. Over equal x-intervals, the output changes by the same constant amount: Δy = m · Δx.
Example: y = 4x + 1. At x = 0, 1, 2, 3: y = 1, 5, 9, 13. First differences: 4, 4, 4. Constant → linear.
Exponential Functions — Multiplicative Change
An exponential function has the form y = ab^x. The variable x appears in the exponent. Over equal x-intervals, the output is multiplied by the same constant factor b: y₂/y₁ = b.
Example: y = 3·2^x. At x = 0, 1, 2, 3: y = 3, 6, 12, 24. Ratios: 2, 2, 2. Constant → exponential.
Neither Linear nor Exponential
If the first differences are not constant and the ratios are not constant, the function is neither linear nor exponential. Common examples include quadratic functions (y = x²), cubic functions (y = x³), and absolute value functions.
Comparison Organizer
| Feature | Linear | Exponential |
|---|---|---|
| Change pattern | Constant difference | Constant ratio |
| Operation | Add or subtract | Multiply or divide |
| Common form | y = mx + b | y = ab^x |
| Graph | Straight line | Curved (exponential) |
| Rate | Constant amount | Constant % or factor |
| Long-term behavior | Steady additive change | Accelerates or decays multiplicatively |
| Example | Adds $20 each month | Increases by 20% each month |
Classifying Functions from Tables
Table Classification — 5-Step Process
- 1
Confirm equal x-intervals
The x-values must increase by the same amount each row. If they do not, you cannot use first differences or ratios.
- 2
Calculate first differences
Subtract each y-value from the next: Δy = y₂ − y₁, y₃ − y₂, etc.
- 3
Calculate common ratios
Divide each y-value by the previous: r = y₂/y₁, y₃/y₂, etc. (Only valid when no y-value is zero.)
- 4
Classify the pattern
Constant difference → linear. Constant nonzero ratio → exponential. Neither constant → neither.
- 5
Verify using another representation
If you have an equation or graph, confirm your classification matches.
Important Cautions
- • Ratios involving zero are undefined — do not force an exponential classification.
- • Do not check only differences or only ratios — check both before concluding.
- • Inconsistent differences and inconsistent ratios → the function is neither linear nor exponential.
Classifying Functions from Equations
Equation Classification — Examples
Critical Distinctions
y = 2x is linear (slope 2). y = 2^x is exponential (x in exponent). y = x² is quadratic (not exponential).
Comparing Graphs
Linear f(x) = 3x + 2 vs Exponential g(x) = 2·(1.8)^x
Both functions start at (0, 2). The exponential overtakes the linear near x ≈ 3.4. For large x, exponential growth dominates.
Reading Graphs for Evidence
A graph gives approximate evidence of crossover and long-term behavior. For exact evidence, use a table or equation. A straight line confirms linear; a curve that accelerates or decays confirms exponential. Note: not every curved graph is exponential — quadratic and other functions also produce curves.
Short-Term vs Long-Term Behavior
Short-Term vs Long-Term: f(x) = 5x + 2 vs g(x) = 2·(1.5)^x
| x | f(x) = 5x + 2 (linear) | g(x) = 2·(1.5)^x (exp.) | Greater |
|---|---|---|---|
| 0 | 2 | 2.0 | Equal |
| 1 | 7 | 3.0 | Linear |
| 2 | 12 | 4.5 | Linear |
| 3 | 17 | 6.8 | Linear |
| 4 | 22 | 10.1 | Linear |
| 5 | 27 | 15.2 | Linear |
| 6 | 32 | 22.8 | Linear |
| 7 | 37 | 34.2 | Linear |
| 8 | 42 | 51.3 | Exponential |
| 10 | 52 | 115.3 | Exponential |
| 15 | 77 | 1303.5 | Exponential |
| 20 | 102 | 14740.0 | Exponential |
The linear function is greater for small x. The exponential overtakes it around x ≈ 7 and grows far faster for large x.
Worked Examples
Identify the function type from the table: x = 0, 1, 2, 3 | y = 5, 8, 11, 14
Step 1: Confirm equal x-intervals: 1, 1, 1. ✓
Step 2: First differences: 8−5=3, 11−8=3, 14−11=3. Constant difference = 3.
Step 3: Ratios: 8/5=1.6, 11/8=1.375, 14/11=1.27. Not constant.
Step 4: Constant difference → LINEAR.
Step 5: Equation form: y = 3x + 5.
Identify the function type from the table: x = 0, 1, 2, 3 | y = 4, 12, 36, 108
Step 1: Confirm equal x-intervals: 1, 1, 1. ✓
Step 2: First differences: 12−4=8, 36−12=24, 108−36=72. Not constant.
Step 3: Ratios: 12/4=3, 36/12=3, 108/36=3. Constant ratio = 3.
Step 4: Constant ratio → EXPONENTIAL.
Step 5: Equation form: y = 4·3^x (a = 4, b = 3).
Identify the function type from the table: x = 0, 1, 2, 3 | y = 1, 4, 9, 16
Step 1: Confirm equal x-intervals: 1, 1, 1. ✓
Step 2: First differences: 4−1=3, 9−4=5, 16−9=7. Not constant.
Step 3: Ratios: 4/1=4, 9/4=2.25, 16/9≈1.78. Not constant.
Step 4: Neither constant differences nor constant ratios.
Step 5: This is y = x² — a quadratic function. NEITHER linear nor exponential.
Compare f(x) = 6x + 10 and g(x) = 5·(2)^x. Identify each type and compare their initial values.
f(x) = 6x + 10: variable x has exponent 1, multiplied by constant 6 → LINEAR.
g(x) = 5·2^x: variable x is in the exponent, base 2 > 1 → EXPONENTIAL GROWTH.
Initial value of f: f(0) = 6(0) + 10 = 10.
Initial value of g: g(0) = 5·2^0 = 5·1 = 5.
f starts higher (10 > 5). However, g grows multiplicatively and will eventually overtake f.
Two functions are graphed. Function A is a straight line through (0, 3) and (4, 11). Function B passes through (0, 3) and (4, 48). Which is linear and which is exponential? Which is greater at x = 6?
Function A: straight line → LINEAR. Slope = (11−3)/(4−0) = 8/4 = 2. Equation: A(x) = 2x + 3.
Function B: curved, passes through (0,3) and (4,48). Check ratio: b^4 = 48/3 = 16 → b = 2. Equation: B(x) = 3·2^x. EXPONENTIAL.
At x = 6: A(6) = 2(6) + 3 = 15.
At x = 6: B(6) = 3·2^6 = 3·64 = 192.
B(6) = 192 >> A(6) = 15. Exponential is far greater at x = 6.
Which model is greater initially: f(x) = 100x + 500 or g(x) = 2·(3)^x?
Initial value of f: f(0) = 100(0) + 500 = 500.
Initial value of g: g(0) = 2·3^0 = 2·1 = 2.
f(0) = 500 >> g(0) = 2.
The linear function f starts much higher initially.
Note: this does not mean f is always greater. Check long-term behavior separately.
For f(x) = 100x + 500 and g(x) = 2·(3)^x, which is greater at x = 10? Estimate the crossover from a table.
f(5) = 1000, g(5) = 2·243 = 486. f > g at x = 5.
f(6) = 1100, g(6) = 2·729 = 1458. g > f at x = 6.
Crossover is between x = 5 and x = 6.
f(10) = 100(10) + 500 = 1500.
g(10) = 2·3^10 = 2·59049 = 118,098.
g(10) >> f(10). Exponential dominates for large x.
A town's population can be modeled by either P(t) = 500t + 2000 or P(t) = 2000·(1.08)^t, where t is years. Which model is more appropriate if the population grows by approximately 8% per year? Justify.
Key information: population grows by approximately 8% per year.
8% per year means each year the population is multiplied by 1.08 — a constant factor.
Constant multiplicative change → EXPONENTIAL model.
P(t) = 2000·(1.08)^t: initial value 2000, growth factor 1.08, growth rate r = 8%.
The linear model P(t) = 500t + 2000 would imply a constant dollar-amount increase of 500 per year, which does not match "8% per year."
Verify: at t = 1, exponential gives 2000·1.08 = 2160 (8% increase). Linear gives 2500 (25% increase). Exponential matches the stated rate.
Common Mistakes
Looking only at whether values increase — both linear and exponential functions can increase.
Check the pattern of change: constant difference (linear) or constant ratio (exponential).
Using ratios when x-intervals are unequal — ratios are only valid for equal x-intervals.
Always confirm equal x-intervals before computing first differences or ratios.
Assuming every curved graph is exponential — quadratic, cubic, and other functions also produce curves.
Confirm with a table or equation. A curve alone is not sufficient evidence of an exponential function.
Confusing constant percent change with constant amount change — "grows 5% per year" is exponential, not linear.
Constant percent → multiplicative → exponential. Constant amount → additive → linear.
Checking only differences and concluding "not linear, must be exponential."
Check both differences and ratios. If neither is constant, the function is neither linear nor exponential.
Ignoring initial values — assuming exponential growth is always greater immediately.
Compare f(0) and g(0). A linear function with a large initial value may be greater than an exponential for small x.
Calling exponential decay linear because values decrease — both can decrease.
Decreasing values can be linear (negative slope) or exponential (0 < b < 1). Check the pattern of change.
Classifying y = x² as exponential because it "grows fast."
y = x² is quadratic — x is the base, not the exponent. Exponential requires the variable in the exponent: y = 2^x.
Classifying y = 2x as exponential because it contains a 2.
y = 2x is linear (slope 2). y = 2^x is exponential (x in the exponent). The position of x matters.
Selecting a model without justification — just guessing "exponential" because the problem mentions growth.
Identify the type of change (additive or multiplicative), write the equation, and verify it matches the data.
Ignoring context and domain — applying an exponential model to a situation with constant dollar amounts.
Read the context carefully. "Adds $50 per month" is linear. "Grows 5% per month" is exponential.
Forgetting that the variable must be in the exponent for an exponential function.
y = ab^x is exponential (x in exponent). y = ax^b is a power function (x is the base). These are different.
Guided Practice
Guided Practice Video: Comparing Linear and Exponential Functions
Watch the guided practice walkthrough for comparing linear and exponential functions, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Classify the table: x = 0, 1, 2, 3 | y = 2, 6, 18, 54. Show your work.
Hint: Calculate first differences and common ratios. Which is constant?
Classify the table: x = 0, 1, 2, 3 | y = 10, 7, 4, 1. Show your work.
Hint: Calculate first differences. Is the slope positive or negative?
Classify the equation: y = 7·(0.5)^x. State the type, initial value, and whether it represents growth or decay.
Hint: Where is the variable x? What is the base? Is the base greater than 1 or between 0 and 1?
Two functions: f(x) = 8x + 3 and g(x) = 3·(2)^x. Which is greater at x = 0? At x = 5? At x = 10?
Hint: Evaluate each function at x = 0, 5, and 10. Compare the outputs.
A savings account earns 3% interest per year. A second account adds $150 per year. Which model is linear and which is exponential? Which account has more money after 20 years if both start at $1,000?
Hint: Identify the type of change for each account. Write the equation for each, then evaluate at t = 20.
Classify: x = 1, 2, 3, 4 | y = 1, 4, 9, 16. Is this linear, exponential, or neither? Justify.
Hint: Check first differences and ratios. Do either remain constant?
Real-World Model Selection
| Context | Change Type | Model | Form |
|---|---|---|---|
| Equal dollar deposit each month | Constant amount (+$200/mo) | Linear | f(t) = 200t + b |
| Investment grows 6% per year | Constant % (×1.06/yr) | Exponential | f(t) = a·(1.06)^t |
| Fixed charge + per-unit fee | Constant amount per unit | Linear | f(x) = mx + b |
| Population multiplies by 1.03/yr | Constant factor (×1.03/yr) | Exponential | f(t) = a·(1.03)^t |
| Fixed-dollar raise each year | Constant amount (+$2,000/yr) | Linear | f(t) = 2000t + a |
| Salary increases 4% per year | Constant % (×1.04/yr) | Exponential | f(t) = a·(1.04)^t |
| Car loses $1,500 value per year | Constant amount (−$1,500/yr) | Linear | f(t) = −1500t + a |
| Car loses 15% of value per year | Constant % (×0.85/yr) | Exponential | f(t) = a·(0.85)^t |