Quick questions on Exponential models for HSC Maths Standard 2

This is the same multiplier as the one used in straight compound interest or declining-balance depreciation.

The graph of $y = a b^x$ (for $a > 0$) approaches the $x$-axis but never touches it. The line $y = 0$ is a horizontal asymptote. Practical interpretation: a decaying quantity gets closer and closer to zero without ever reaching exactly zero.

If $y = a b^x$ and you know $y$, $a$ and $b$ and want $x$:

:::worked Worked example ### Compound interest growth

Sydney's population in $2021$ was about $5.26$ million. If growth continues at $1.3\%$ per year (ABS estimate, 2021 census period), the model is $P = 5.26 (1.013)^t$ million.

A laptop bought for \1800$depreciates at$20\%$per year. Model:$V = 1800 (0.80)^t$.

$12\%$ decay is base $0.88$. Easy to write $1.12$ by reflex.

At $x = 0$, the function equals $a$. Use this to read off the starting amount from the model.

Standard 2 expects you to take logs and rearrange. Use $\log$ or $\ln$, your choice.