Trigonometric functions and their graphsIB MYP Maths Extended: Subtopic test

1. 1
   A function is defined by $y=3\sin(2x)+1$ for $0^\circ\le x\le360^\circ$.

   (a)

   What is the period of the function?

   [1 mark]

   • A$90^\circ$
   • B$180^\circ$
   • C$360^\circ$
   • D$720^\circ$

   (b)

   What is the greatest value of $y$?

   [1 mark]

   • A$3$
   • B$7$
   • C$4$
   • D$6$

   (c)

   Write down the equation of the midline of the graph and its amplitude.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A student uses technology to draw the graph of $y=\sin(bx)$ for $0^\circ\le x\le360^\circ$ and records the period for several values of $b$: $b=1$ gives $360^\circ$, $b=2$ gives $180^\circ$, $b=3$ gives $120^\circ$ and $b=4$ gives $90^\circ$.

   (a)

   What is the period of $y=\sin(6x)$?

   [1 mark]

   • A$60^\circ$
   • B$6^\circ$
   • C$2160^\circ$
   • D$30^\circ$

   (b)

   Which rule gives the period of $y=\sin(bx)$ for every value of $b$ in the data?

   [1 mark]

   • A$360b$
   • B$360+b$
   • C$360-b$
   • D$\frac{360}{b}$

   (c)

   The student now graphs $y=\sin(bx)$ and finds that the period is $45^\circ$. Use the rule to find $b$, and verify your answer by checking that the rule still gives the recorded period when $b=4$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The depth of water, $h$ metres, in a harbour in Lisbon is modelled by $h=2\sin(30t)+5$, where $t$ is the time in hours after midnight and $0\le t\le12$. The angle $30t$ is measured in degrees.

   (a)

   State the greatest depth, the least depth and the time between two consecutive high tides.

   [3 marks]

   (b)

   A boat can enter the harbour only when the depth is at least 6 m. Find the times at which the depth is exactly 6 m.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The number of hours of daylight, $d$, in a northern European city on day $n$ of the year ($n=1$ is 1 January) is modelled by $d=6\sin\left(\frac{360}{365}(n-80)\right)+12$, where the angle is in degrees. The city's records show that its shortest day has 6.4 hours of daylight and its longest day has 17.6 hours.

   (a)

   (i) State the greatest and least number of daylight hours that the model gives.
   (ii) Find the model's value of $d$ on day 172, to 1 decimal place.
   (iii) Find, to the nearest day, the days in the first year on which the model gives exactly 15 hours of daylight.

   [6 marks]

   (b)

   Use the city's records to evaluate how accurate the model is, and suggest an improvement to the model.

   [6 marks]

   Total for question 4: 12 marks

End of questions