The In-Betweeners

Learning Focus

Interpolate values between known points for linear and exponential functions.

Big idea

To find a value between two known inputs:

• Linear: outputs change by a constant amount. The in-between value for a midpoint is the average of the two outputs.
• Exponential: outputs change by a constant factor. The in-between value for a midpoint is the geometric mean $\sqrt{y_1\cdot y_2}$.

For exponentials, find the growth factor $b$ per unit input and multiply (or use $f(x)=a\,b^x$).

Worked example

Known: $f(0)=3$ and $f(2)=12$. Find $f(1)$.

• Linear: $f(1)=\dfrac{3+12}{2}=7.5$
• Exponential: $f(1)=\sqrt{3\cdot12}=6$

Exponential check: $b^2=\dfrac{12}{3}=4\Rightarrow b=2$, so $f(x)=3\cdot2^x$ and $f(1)=6$ ✓.

Takeaways

• Linear: add equal amounts; exponential: multiply by equal factors.
• Midpoint of an exponential is the geometric mean, not the average.
• Write the model $a\,b^x$ to find any in-between value.