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Compare Linear and
Exponential Growth

Learn how linear functions grow by equal differences and exponential functions grow by equal factors over equal intervals.

Algebra 2

What Is Growth in a Function?

When we talk about the growth of a function, we are asking: how does the output change as the input increases by the same amount each time? Two very different patterns dominate algebra:

Linear Growth
The output increases (or decreases) by a constant amount for each unit step in \( x \).
Exponential Growth
The output is multiplied by a constant factor for each unit step in \( x \).

Linear Functions: Equal Differences

A linear function has the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept. The defining property:

Example: if \( f(x) = 3x + 2 \), then every time \( x \) increases by 1, the output increases by 3: the same amount every single time.

Exponential Functions: Equal Factors

An exponential function has the form \( g(x) = a \cdot b^x \), where \( a \neq 0 \) and \( b > 0,\; b \neq 1 \). The defining property:

Example: if \( g(x) = 5 \cdot 2^x \), then every time \( x \) increases by 1, the output is multiplied by 2.

See It in Numbers

Linear: \( f(x) = 3x + 1 \)
\( f(0) = 1 \)
\( f(1) = 4 \)   diff = +3
\( f(2) = 7 \)   diff = +3
\( f(3) = 10 \)   diff = +3
\( f(4) = 13 \)   diff = +3
Exponential: \( g(x) = 2^x \)
\( g(0) = 1 \)
\( g(1) = 2 \)   ratio = ×2
\( g(2) = 4 \)   ratio = ×2
\( g(3) = 8 \)   ratio = ×2
\( g(4) = 16 \)   ratio = ×2

Visualizing Both Curves

\(x\)
\(y\)
 0 1 2 3 4 0 4 8 12 16
\( f(x) = 3x + 1 \)
\( g(x) = 2^x \)

Proving Equal Differences

Let \( f(x) = mx + b \). Pick any starting value \( x \) and a fixed interval \( h \). Compute the difference:

Proving Equal Factors

Let \( g(x) = a \cdot b^x \). Pick any starting value \( x \) and a fixed interval \( h \). Compute the ratio:

Differences vs. Ratios

Check Differences
Subtract consecutive outputs: \( f(x+1) - f(x) \). If the result is always the same, the function is linear.
Check Ratios
Divide consecutive outputs: \( \dfrac{g(x+1)}{g(x)} \). If the result is always the same, the function is exponential.

How to Identify the Type

1
Confirm equal spacing. Make sure the \( x \)-values in the table are evenly spaced (e.g., 0, 1, 2, 3 or 0, 2, 4, 6).
2
Compute first differences. Subtract each output from the next: \( y_2 - y_1,\; y_3 - y_2, \ldots \) If all differences are equal, the function is linear.
3
Compute consecutive ratios. Divide each output by the previous: \( \dfrac{y_2}{y_1},\; \dfrac{y_3}{y_2}, \ldots \) If all ratios are equal, the function is exponential.
4
Write the rule. For a linear function, the slope is \( m = \frac{\text{common difference}}{\Delta x} \). For an exponential function, the base is \( b = (\text{common ratio})^{1/\Delta x} \).

Example 1: Classify from a Table

\( x \) 0 1 2 3 4
\( y \) 5 15 45 135 405

Example 2: Classify from a Table

\( x \) 0 2 4 6 8
\( y \) 7 19 31 43 55

Example 3: Real-World Context

A bank account starts with $200 and earns 5% interest per year. A savings jar starts with $200 and receives $30 per year. Which is linear? Which is exponential?

Savings Jar
\( S(t) = 200 + 30t \)
Adding the same $30 each year: linear.
Bank Account
\( B(t) = 200 \cdot 1.05^t \)
Multiplying by 1.05 each year: exponential.

Why Exponential Always Wins

0 10 20 30 40 50
Years
 0 600 1200 1800 2400 Jar (linear) Bank (exponential)

Common Mistakes

✕ Confusing Differences and Ratios
Students check differences for an exponential table and see they are "increasing," concluding the growth rate is "changing." Instead, check ratios for exponential.
✕ Unequal Intervals
The equal-difference / equal-ratio test only works when the \( x \)-values are equally spaced. If \( x = 1, 2, 4, 7 \), you cannot directly compare consecutive outputs.
✕ Forgetting \( a \neq 0 \)
In \( g(x) = a \cdot b^x \), if \( a = 0 \) the function is just \( y = 0 \): not exponential. The initial value \( a \) must be nonzero.

Summary Comparison

Linear Exponential
General Form \( f(x) = mx + b \) \( g(x) = a \cdot b^x \)
Growth Pattern Add constant \( m \) Multiply by constant \( b \)
Key Test Equal differences Equal ratios
Graph Shape Straight line Curve (concave up if \( b > 1 \))
Long-Term Behavior Steady, predictable Accelerating, eventually dominates
Real Example $30/year savings 5% annual interest

Question 1 of 5

A function has the values \( f(0)=4,\; f(1)=12,\; f(2)=36,\; f(3)=108 \). What type of function is this?

Question 2 of 5

For the linear function \( f(x) = -4x + 10 \), what is \( f(5) - f(3) \)?

Question 3 of 5

An exponential function satisfies \( g(2)=50 \) and \( g(4)=200 \). What is the common ratio \( b \) (per unit increase in \( x \))?

Question 4 of 5

Which statement best explains why a linear function has equal differences over equal intervals?

Question 5 of 5

A population doubles every 3 years. If the current population is 500, which function models this growth?