Functions and graphsIB MYP Maths Extended: Topic test

1. 1
   A function is defined by $f(x)=x^{2}-4x$ for all real values of $x$.

   (a)

   Evaluate $f(-2)$.

   [1 mark]

   • A$-4$
   • B$12$
   • C$0$
   • D$-12$

   (b)

   Which values of $x$ satisfy $f(x)=5$?

   [1 mark]

   • A$x=-5$ or $x=1$
   • B$x=5$ or $x=1$
   • C$x=-5$ or $x=-1$
   • D$x=5$ or $x=-1$

   (c)

   State the range of $f$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A straight line passes through the points $P(-1,5)$ and $Q(3,-3)$.

   (a)

   What is the gradient of the line $PQ$?

   [1 mark]

   • A$-2$
   • B$2$
   • C$-\frac12$
   • D$\frac12$

   (b)

   What is the gradient of a line perpendicular to $PQ$?

   [1 mark]

   • A$-2$
   • B$2$
   • C$\frac12$
   • D$-\frac12$

   (c)

   Find the equation of the line $PQ$ in the form $y=mx+c$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A zip wire in an adventure park in Cape Town runs in a straight line from platform $A(-3,2)$ to platform $B(5,-4)$ on a grid where one unit represents $10$ m.

   (a)

   Find the length of the zip wire in metres.

   [3 marks]

   (b)

   A support cable runs from the midpoint $M$ of $AB$ along the line through $M$ that is perpendicular to $AB$. Find the equation of this line.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two candles are lit at the same time. Candle A is $30$ cm tall and burns down at a steady $2.5$ cm each hour. Candle B is $24$ cm tall and burns down at a steady $1.5$ cm each hour. After $t$ hours the heights of the candles are $h_A$ cm and $h_B$ cm.

   (a)

   (i) Write an equation for $h_A$ and an equation for $h_B$ in terms of $t$. [2] (ii) Find the time at which the two candles are the same height, and that height. [3] (iii) Describe which candle is taller before and after this time. [1]

   [6 marks]

   (b)

   State the domain and range of $h_A$ in this context. A shop claims that Candle B burns for $25\%$ longer than Candle A. Show whether the claim is correct.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A function $f$ has a graph with a single turning point, which is a maximum at $(3,5)$.

   (a)

   The graph of $y=f(x)+2$ is drawn. What are the coordinates of its maximum point?

   [1 mark]

   • A$(1,5)$
   • B$(3,3)$
   • C$(5,5)$
   • D$(3,7)$

   (b)

   The graph of $y=f(x-2)$ is drawn. What are the coordinates of its maximum point?

   [1 mark]

   • A$(5,5)$
   • B$(1,5)$
   • C$(3,7)$
   • D$(3,3)$

   (c)

   The graph of $y=-f(x)$ is drawn. Write down the coordinates of its turning point and state whether it is a maximum or a minimum.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A hospital in Lima uses a medical tracer. The activity $A$ of the tracer (in MBq) $t$ hours after it is injected is modelled by $A=800\times0.8^{t}$.

   (a)

   Find the activity of the tracer $2$ hours after injection.

   [1 mark]

   • A$640$ MBq
   • B$480$ MBq
   • C$512$ MBq
   • D$160$ MBq

   (b)

   Which statement about the graph of $A=800\times0.8^{t}$ is correct?

   [1 mark]

   • AIt crosses the $t$-axis at $t=10$.
   • BIt gets closer and closer to the $t$-axis but never reaches it.
   • CIt is a straight line through $(0,800)$.
   • DIt increases as $t$ increases.

   (c)

   Find the smallest whole number of hours after which the activity is below $100$ MBq. Show your working.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Consider the cubic function $f(x)=x(x+3)(x-2)$.

   (a)

   Write down the $x$-intercepts of the graph of $y=f(x)$ and describe what happens to $f(x)$ as $x$ becomes very large and positive, and very large and negative.

   [3 marks]

   (b)

   Use graphing technology to find the coordinates of the local maximum point and the local minimum point of $y=f(x)$, giving each coordinate to $2$ decimal places.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A school tuck shop in Lagos makes $x$ sandwiches and $y$ wraps each day. It can make at most $40$ items in total. The ingredients cost NGN $300$ per sandwich and NGN $500$ per wrap, and the daily ingredients budget is NGN $15\,000$. The profit is NGN $200$ on each sandwich and NGN $300$ on each wrap.

   (a)

   Write down the inequalities that the constraints give, and find the coordinates of all the vertices of the feasible region.

   [6 marks]

   (b)

   Find the greatest daily profit and the numbers of sandwiches and wraps that give it. The shop then lowers its price so that the profit on a wrap falls to NGN $100$. Find the new best numbers of sandwiches and wraps and say whether the plan has changed.

   [6 marks]

   Total for question 8: 12 marks

End of questions