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Linear & Exponential
Growth

Master the difference between constant rate and constant ratio growth: the two most tested function families on the SAT.

SAT Math

What Does Growth Mean?

In math, growth describes how a quantity changes over time or across inputs. On the SAT, you will see two main patterns:

Linear Growth
The quantity increases (or decreases) by the same amount each step: a constant difference.
Exponential Growth
The quantity increases (or decreases) by the same factor each step: a constant ratio.

The Linear Function

A linear function has the form:

$$f(x) = mx + b$$

\( m \) : Slope
The constant rate of change: how much \( f(x) \) changes for each unit increase in \( x \).
\( b \) : y-intercept
The starting value: the output when \( x = 0 \).

The Exponential Function

An exponential function has the form:

$$f(x) = a \cdot b^{\,x}$$

\( a \) : Initial Value
The output when \( x = 0 \). Must be non-zero.
\( b \) : Growth Factor
The constant multiplier per unit step. Growth when \( b > 1 \); decay when \( 0 < b < 1 \).

Seeing the Difference

0 1 2 3 4 5 0 5 10 15 20 25 \(f(x)\) \(x\) Linear: \(2x + 1\) Exponential: \(2^x\)

Constant Difference vs. Constant Ratio

Linear Exponential
General Form \( mx + b \) \( a \cdot b^x \)
Key Pattern Add a constant Multiply by a constant
Table Test Equal \( \Delta y \) for equal \( \Delta x \) Equal \( \tfrac{y_2}{y_1} \) for equal \( \Delta x \)
Graph Shape Straight line Curve (J-shape or decay)
Real-World Example Hourly wage Compound interest

Growth vs. Decay

Exponential Growth
\( b > 1 \): the function increases. Example: \( f(x) = 3 \cdot 2^x \). The factor \( 2 \) doubles the value each step.
Exponential Decay
\( 0 < b < 1 \): the function decreases. Example: \( f(x) = 100 \cdot (0.8)^x \). Each step retains 80% of the previous value.

How to Identify the Model

1
Check for a table or word cue. Look for phrases like "increases by \( 5 \) each year" (linear) or "doubles every \( 3 \) hours" (exponential).
2
Compute consecutive differences. Subtract \( y_2 - y_1 \), \( y_3 - y_2 \), etc. If they are equal, the model is linear.
3
Compute consecutive ratios. Divide \( \tfrac{y_2}{y_1} \), \( \tfrac{y_3}{y_2} \), etc. If they are equal, the model is exponential.
4
Write the equation. For linear: \( y = mx + b \). For exponential: \( y = a \cdot b^x \). Plug in a known point to verify.

Example 1: Linear Table

\( x \) 0 1 2 3 4
\( f(x) \) 5 8 11 14 17

Example 2: Exponential Table

\( x \) 0 1 2 3 4
\( g(x) \) 4 12 36 108 324

Example 3: Percent Change

Problem: A car purchased for $25,000 depreciates by 15% per year. Write a function \( V(t) \) for the car's value after \( t \) years.

1
Identify the model. "Depreciates by 15% per year" means a constant percent decrease: exponential decay.
2
Find the parameters. Initial value \( a = 25{,}000 \). Decay factor \( b = 1 - 0.15 = 0.85 \).
3
Write the equation. \( V(t) = 25{,}000 \cdot (0.85)^t \).

Interpreting Parameters on the SAT

SAT questions often ask: "What does the number \( k \) represent in the equation?" Match each parameter to its real-world meaning:

Linear \( y = mx + b \)
\( m \): rate of change per unit time.
\( b \): starting value at time 0.
Exponential \( y = a \cdot b^t \)
\( a \): initial amount.
\( b \): growth/decay factor (not the percent itself).

The Compound Growth Formula

$$A = P\left(1 + \frac{r}{n}\right)^{nt}$$

\( P \)
Principal (initial amount)
\( r \)
Annual rate (as a decimal)
\( n \)
Compounding periods per year
\( t \)
Time in years

When Do They Intersect?

Time Value Linear Exponential Intersection Linear ahead here Exponential overtakes

Common Mistakes

✕ Confusing Factor and Rate
A growth factor of \( 1.05 \) means a 5% increase, not a 105% increase. Subtract 1 to find the rate.
✕ Adding Instead of Multiplying
For exponential growth, each step multiplies by \( b \). Writing \( a + bx \) creates a linear model instead.
✕ Wrong Base for Decay
A 20% decrease uses \( b = 0.80 \), not \( b = 0.20 \). The base represents what remains, not what is lost.

Summary Cheat Sheet

Linear Exponential
Equation \( y = mx + b \) \( y = a \cdot b^x \)
Change Pattern Add \( m \) per step Multiply by \( b \) per step
Graph Straight line Curve
Growth Cue "increases by 10 each..." "increases by 10% each..."
Decay Cue \( m < 0 \) \( 0 < b < 1 \)
Long-Term Winner Always overtaken Always overtakes

Question 1 of 5

A population of bacteria triples every hour. If the initial population is 200, which function models the population \( P \) after \( t \) hours?

Question 2 of 5

The table shows values of a function \( f \).

\( x \) 0123
\( f(x) \) 61854162

What is \( f(5) \)?

Question 3 of 5

A savings account earns 4% interest compounded annually. If $1,000 is deposited, which expression gives the account balance after \( t \) years?

Question 4 of 5

A car loses 12% of its value each year. If the car is currently worth $20,000, what will it be worth in 2 years, rounded to the nearest dollar?

Question 5 of 5

A gym membership costs $50 to join and $30 per month. A second gym charges no joining fee but $35 per month. After how many months will the total cost be the same?