Exponential Models Practice

These questions follow The Exponential Function — recognising the family, reading its properties, and putting it to work on growth and decay. Round final answers sensibly and keep the model exact until the last step.

Recognising the family

1. A function’s outputs at $x=0,1,2,3$ are $3,6,12,24$. Explain how you know the function is exponential rather than linear or quadratic, and write its equation.
2. For $f(x)=5^x$: state the $y$-intercept, domain, range, equation of the asymptote, and whether the function increases or decreases.
3. How is $g(x)={\left(\frac{1}{2}\right)}^x$ related to $f(x)=2^x$? Give both an exponent-law answer and a transformation answer.

Answer 1

First differences are $3,6,12$ — not constant, so not linear; nor do those differences change steadily, so not quadratic. But each output is double the last: a constant ratio, the exponential signature. Equation: $f(x)=3\cdot 2^x$ — initial value 3, ratio 2.

Answer 2

$y$-intercept 1 (since $5^0=1$); domain all real numbers; range $y>0$; asymptote $y=0$; increasing everywhere, since the base exceeds 1.

Answer 3

By the laws, ${\left(\frac{1}{2}\right)}^x=2^{-x}$. As a transformation, that is $f(x)=2^x$ reflected in the $y$-axis. One function, two descriptions — and its graph decreases, as any base between 0 and 1 must.

Growth and decay

4. A town’s population is modelled by $P(t)=2000(1.03{)}^t$, with $t$ in years. What do the 2000 and the 1.03 each mean, and what is the population after 10 years?
5. A patient receives 80 mg of a medication with a half-life of 6 hours, so $M(t)=80{\left(\frac{1}{2}\right)}^{t/6}$. How much remains after 15 hours?
6. A cooling drink follows $T(x) = 60\left(\frac{1}{2}\right)^{x/30}
   • 20$,with$T$in°Cand$x$inminutes.Statetheinitialtemperature,themeaningofthe$+20$, and the temperature after one hour.
7. A ball dropped from 2 m rebounds to 60% of its previous height each bounce. Find the height of the fourth bounce.

Answer 4

2000 is the starting population; 1.03 is the annual multiplier — each year keeps 100% and adds 3%. After 10 years: $P(10)=2000(1.03{)}^{10}\approx 2688$ people.

Answer 5

Fifteen hours is $\frac{15}{6}=2.5$ half-lives, so $M(15)=80{\left(\frac{1}{2}\right)}^{2.5}\approx 14.1$ mg. Sanity check: two half-lives leave 20 mg, three leave 10, and 14.1 sits between them. ✓

Answer 6

At $x=0$: $T=60+20=80$°C. The $+20$ is the room’s temperature — the asymptote the drink cools toward but never quite reaches. After one hour, $x=60$ is two half-lives of the gap: $T(60)=60\left(\frac{1}{4}\right)+20=35$°C.

Answer 7

Each bounce multiplies by 0.6: $h=2(0.6{)}^4=2(0.1296)\approx 0.26$ m. The model’s domain is honest here too — bounce number is a whole number, so this is really a geometric sequence wearing a decay costume.

Build your own

8. An exponential function has $y$-intercept 5 and horizontal asymptote $y=3$. Give one possible equation, and explain why more than one function fits.

Answer 8

The asymptote forces the form $y=a\cdot b^x+3$; the intercept forces $a+3=5$, so $a=2$. Any allowed base works: $y=2(2{)}^x+3$ and $y=2(5{)}^x+3$ both satisfy the clues, and $y=2(2{)}^{-x}+3$ sneaks in as a decreasing option. Two properties pin down two letters, and the base was never mentioned — so it stays free. Compare candidates in Using Desmos.