Unit 7: Exponential and General Functions

Topic 2

Comparing Linear and Exponential Growth

Linear and exponential functions both model change over time, but in very different ways:

\text{Linear: } y = mx + b \qquad \text{Exponential: } y = a b^x

A linear function increases (or decreases) by a constant amount with each step in

x

. An exponential function increases (or decreases) by a constant factor (percentage) with each step in

x

\begin{cases} \text{Linear: } \Delta y = \text{constant} \\ \text{Exponential: } \dfrac{y_2}{y_1} = \text{constant ratio} \end{cases}

Early on, a linear function might appear to grow faster, but eventually an exponential function surpasses it because it compounds multiplicatively. For example,

y = 2x + 5

starts out above

y = 2^x

, but

2^x

overtakes it between

x = 3

and

x = 4

When to Use Each Model

Use a linear model when a fixed amount is added or subtracted (e.g., salary increases by $1000 each year).
Use an exponential model when change is proportional to the current value (e.g., population grows 5% per year).

Graphically:

Linear graphs are straight lines.
Exponential graphs curve upward (growth) or downward (decay).

y = 2x + 5 (linear)
y = 2^x (exponential)

Core Skills

Identify whether change is additive (linear) or multiplicative (exponential).
Write equations for both models given a situation or data.
Compare values over time to see when exponential overtakes linear.
Interpret meaning of slope ( $m$ ) and growth factor ( $b$ ).
Recognize patterns in tables and graphs that distinguish the two.

Example 1: Distinguishing Linear vs. Exponential

A quantity increases by 10 each year.

y = 50 + 10x

Constant difference → linear. Another quantity increases by 10% each year.

y = 50(1.10)^x

Constant ratio → exponential. Answer: First is linear, second is exponential.

Example 2: Comparing Growth Over Time

Compare

L(x) = 100 + 20x

and

E(x) = 100(1.15)^x.

Compute a few values:

\begin{array}{c|ccccc} x & 0 & 2 & 4 & 6 & 8 \\ \hline L(x) & 100 & 140 & 180 & 220 & 260 \\ E(x) & 100 & 132 & 175 & 231 & 306 \end{array}

At first, linear grows faster, but around

x = 5

, exponential surpasses it. Interpretation: Exponential growth dominates over time.

Example 3: Real-World Comparison

A worker earns $30,000 and gets a $2,000 raise yearly (linear):

S_L(t) = 30{,}000 + 2000t

A startup grows revenue by 10% annually (exponential):

S_E(t) = 30{,}000(1.10)^t

After 5 years:

S_L(5) = 40{,}000, \quad S_E(5) = 30{,}000(1.10)^5 = 48{,}315.

The exponential model surpasses the linear one. Answer: Exponential wins in the long term.

Example 4: Identifying Model Type from Data

\begin{array}{c|ccccc} x & 0 & 1 & 2 & 3 & 4 \\ \hline y & 10 & 20 & 30 & 40 & 50 \end{array}

Constant difference of +10 → linear.

\begin{array}{c|ccccc} x & 0 & 1 & 2 & 3 & 4 \\ \hline y & 10 & 15 & 22.5 & 33.75 & 50.6 \end{array}

Constant ratio (multiply by 1.5) → exponential.

Example 5: Crossover Point (When They Are Equal)

Solve

L(x) = E(x)

for

100 + 20x = 100(1.15)^x.

This cannot be solved algebraically without a calculator, but graphically they intersect near

x \approx 5.2.

Beyond that point, exponential dominates.

Key Takeaways

Linear: constant difference; Exponential: constant ratio.
Linear growth is steady; exponential growth accelerates.
On a graph, exponential curves eventually overtake linear lines.
In data, check differences vs. ratios to identify the model.
On the SAT, expect comparison questions between linear and exponential models.