Another Day

Learning Focus

Synthesize function features to compare linear and exponential functions and make predictions.

Big idea

To compare two functions, line up their features:

• Starting values ($y$-intercepts)
• Rate of change (constant vs. changing)
• Intercepts, increasing/decreasing, asymptotes
• Where one overtakes the other (compare outputs in a table)

Exponential growth with $b>1$ eventually surpasses any linear function, even if it starts lower.

Worked example

Compare $f(x)=2x+10$ and $g(x)=3\cdot2^x$.

| $x$ | $f$ | $g$ | |---|---|---| | 0 | 10 | 3 | | 2 | 14 | 12 | | 3 | 16 | 24 | | 4 | 18 | 48 |

$g$ starts lower but overtakes $f$ between $x=2$ and $x=3$. Both are increasing for $x\ge0$; $f$ adds $2$ each step, $g$ doubles.

Takeaways

• Use tables to find where one function overtakes another.
• Linear: add. Exponential: multiply.
• Describe both starting values and long-term behavior.