Linear vs Exponential on the SAT
Growth Patterns That Matter
Build equations from context, spot patterns fast, and practice with intent.
Why the SAT Emphasizes Linear vs Exponential
Linear growth adds the same amount each step. Exponential growth multiplies by the same factor each step. The SAT expects you to recognize the difference quickly from a table, graph, or equation.
This lesson shows a simple test for each type, constant differences for linear and constant ratios for exponential. You will also see how to explain the difference in words so your answer matches the context.
A Simple Definition Unlocks Linear vs Exponential
In a linear pattern, the change in $y$ is constant when $x$ increases by $1$. In an exponential pattern, the ratio of consecutive $y$ values stays constant.
If the table is small, both patterns can look similar at first. Check more than one step and be careful when the starting value is $0$, because ratios are not defined there.
Work Through Linear vs Exponential Step by Step
The sequence is $2, 5, 8, 11, \dots$. Is it linear or exponential, and which model fits it?
Determine whether the pattern is exponential by checking ratios between successive values.
Find the first difference to check for constant change.
Find the second difference to confirm the pattern.
Find the third difference to confirm the pattern.
Find the first ratio to check for constant growth.
Find the second ratio to confirm the growth factor.
Find the third ratio to confirm the pattern.
Conclude it is linear because the differences stay constant.
Conclude it is exponential because the ratios stay constant.
Use Desmos to Check Linear vs Exponential
The sequence is $2, 5, 8, 11, \dots$. Is it linear or exponential, and which model fits it?
Desmos can graph both models and make the contrast visible. Exponential curves rise slowly at first and then shoot upward, while linear graphs are straight lines.
y = 3x + 2
y = 2(2^x)
Algebra is faster for table questions. Desmos is helpful when a graph comparison is needed.
Desmos is faster for a visual comparison of growth. Algebra is faster when a short table makes the pattern obvious.
Expert move: Graph the key functions together and use intercepts or a table to compare how values change; Desmos makes the shape (linear vs exponential, shifts, stretches) obvious.
Reminder: Desmos confirms the picture, but you still have to interpret axes, units, and context.
- Desmos features used: graphing, visual comparison.
- Common mistake: thinking any curve is exponential, even when differences are constant.
Practice Linear vs Exponential with SAT-Style Questions
Identify the growth type and interpret models.
Which equation represents exponential growth?
A population doubles every year. Which model fits?
The table shows values: when is . Which model fits?
Which statement best describes exponential growth compared to linear growth?
Key Takeaways to Remember for Linear vs Exponential
- Linear models have constant differences, exponential models have constant ratios.
- Look for a variable in the exponent to identify exponential functions.
- Desmos makes the contrast between curves and lines obvious.
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