Exponential functionsIB MYP Maths Extended: Subtopic test

1. 1
   Consider the exponential function $f(x)=3^x$.

   (a)

   Find the value of $f(-2)$.

   [1 mark]

   • A$-9$
   • B$\frac{1}{9}$
   • C$9$
   • D$-6$

   (b)

   What is the equation of the horizontal asymptote of the graph of $y=3^x$?

   [1 mark]

   • A$x=0$
   • B$y=1$
   • C$y=0$
   • D$y=3$

   (c)

   Solve $3^x=\frac{1}{27}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A car is bought for $\$24\,000$. Its value $V$ dollars after $t$ years is modelled by $V=24\,000\times0.85^t$.

   (a)

   Find the value of the car after 2 years.

   [1 mark]

   • A$\$17\,340$
   • B$\$540$
   • C$\$20\,400$
   • D$\$16\,800$

   (b)

   After how many whole years is the car first worth less than half of its original price ($\$12\,000$)?

   [1 mark]

   • A3 years
   • B4 years
   • C7 years
   • D5 years

   (c)

   Write down the equation of the horizontal asymptote of the graph of $V$ against $t$, and explain what it tells you about the value of the car.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A biologist counts the bacteria in a culture each hour. At the start there are 500 bacteria; after 1 hour there are 600, after 2 hours 720 and after 3 hours 864.

   (a)

   Show that the growth is exponential and find a model of the form $N=a\times b^t$ for the number of bacteria $N$ after $t$ hours.

   [3 marks]

   (b)

   Use your model to find the number of bacteria after 10 hours, and use technology to find how long it takes for the number of bacteria to reach 5000. Give each answer to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Mei invests $\$5000$ in an account that pays 3.5% compound interest per year. Omar invests $\$5000$ in an account that pays 4% simple interest per year, which means 4% of the original $\$5000$ is added each year. Neither of them withdraws any money.

   (a)

   (i) Write down a formula for the value $A$ of Mei's account after $n$ years.
   (ii) Find the value of Mei's account after 8 years.
   (iii) Use technology to find the number of whole years before Mei's account first reaches $\$7000$.

   [6 marks]

   (b)

   The two friends plan to leave their money invested for 20 years. Justify which account is better for a 20-year investment, and find the year in which Mei's account first holds more than Omar's. Use technology where helpful.

   [6 marks]

   Total for question 4: 12 marks

End of questions