Algebra
Exponential functions
Comparing linear and exponential models
20 practice questions
2 video lessons
Theory + worked examples
Theory
Linear and exponential models grow differently:
- Linear \(y=mx+b\): adds a constant each step (constant difference).
- Exponential \(y=a\cdot b^x\): multiplies each step (constant ratio).
Exponential growth eventually overtakes any linear growth.
The exponential overtakes the line.
Telling them apart.
The distinguishing test:
\[\Delta\text{ constant}:\text{linear},\quad \text{ratio constant}:\text{exponential}\]
Check differences and ratios in a table.
How to compare
- Compute the differences between terms.
- Compute the ratios between terms.
- Constant difference \(\Rightarrow\) linear.
- Constant ratio \(\Rightarrow\) exponential.
Example 1 β Constant difference
Which model fits \(3,5,7,9\)?
Solution
Each term is \(+2\) β a constant difference.
| \(\Rightarrow\) | \(\text{linear}\) |
Example 2 β Constant ratio
Which model fits \(2,4,8,16\)?
Solution
Each term is \(\times2\) β a constant ratio.
| \(\Rightarrow\) | \(\text{exponential}\) |
Example 3 β Which grows faster?
Long term, which grows faster: \(y=2x\) or \(y=2^x\)?
Solution
Exponential eventually overtakes any line.
| \(2^x\) | \(\gg\) | \(2x\ \text{for large } x\) |
Example 4 β Recognize the pattern
How do you tell linear from exponential in a table?
Solution
Constant difference is linear; constant ratio is exponential.
Common pitfalls
Constant difference is linear; constant ratio is exponential.
Exponential eventually wins over linear growth.
Check both differences and ratios.
Frequently asked questions
How is a linear model different from exponential?
Linear adds a constant; exponential multiplies.
What indicates a linear model?
A constant difference between terms.
What indicates an exponential model?
A constant ratio between terms.
Which grows faster in the long run?
Exponential β it overtakes any line.
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