2.3 Graphs of functions and sketchingIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   A student uses a GDC to graph $f(x)=x^2-4x-5$ and then transfers the graph to paper.

   (a)

   An exam question asks the student to sketch the graph of $f$. Which of the following best describes a sketch?

   [1 mark]

   • AAn accurate plot on graph paper, drawn to scale with a pencil and ruler
   • BA table of values with no graph
   • CA set of plotted points joined by straight lines
   • DA graph showing the general shape and relevant features, such as intercepts and the vertex, with the axes and key points labelled

   (b)

   Use your GDC to find the $x$-intercepts of the graph of $f$.

   [1 mark]

   • A$x=1$ and $x=-5$
   • B$x=-1$ and $x=-5$
   • C$x=-1$ and $x=5$
   • D$x=0$ and $x=4$

   (c)

   Find the coordinates of the $y$-intercept and of the vertex of the graph of $f$. These are the points to label on a sketch.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A small factory makes $x$ hundred items per week, where $0\le x\le15$. The weekly revenue is $R(x)=30x$ and the weekly cost is $C(x)=2x^2+50$, both in thousands of AED. The weekly profit is $P(x)=R(x)-C(x)$.

   (a)

   Which of the following is the expression for $P(x)$?

   [1 mark]

   • A$-2x^2+30x+50$
   • B$-2x^2+30x-50$
   • C$2x^2-30x+50$
   • D$2x^2+30x+50$

   (b)

   Use your GDC to find the maximum weekly profit.

   [1 mark]

   • A$62.5$ thousand AED
   • B$7.5$ thousand AED
   • C$225$ thousand AED
   • D$50$ thousand AED

   (c)

   Use your GDC to find the values of $x$ for which the factory breaks even, that is $P(x)=0$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A cup of tea cools in a room. Its temperature, $T$ °C, at $t$ minutes after it is poured is modelled by $T(t)=20+70\times0.9^t$, for $0\le t\le30$.

   (a)

   Use your GDC where appropriate.
   (i) Write down $T(0)$.
   (ii) Find the temperature of the tea after 10 minutes.
   (iii) Find the time at which the temperature of the tea is 50 °C.

   [3 marks]

   (b)

   Describe the key features that should be labelled on a sketch of the graph of $T$ for $0\le t\le30$: the shape of the curve, the vertical intercept, the horizontal asymptote and the coordinates of the end point.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A bank offers two savings accounts, for $0\le t\le20$ years. The value of account A is $A(t)=1000\times1.05^t$ USD and the value of account B is $B(t)=800+90t$ USD. Use your GDC where appropriate.

   (a)

   (i) Find the values of $t$ for which $A(t)=B(t)$.
   (ii) Hence state the values of $t$ for which account B is worth more than account A.
   (iii) Find the difference in value between the two accounts after 20 years.

   [6 marks]

   (b)

   Let $D(t)=A(t)-B(t)$.
   (i) Find $D(0)$ and interpret its value.
   (ii) Use your GDC to find the minimum value of $D(t)$ and the value of $t$ at which it occurs.
   (iii) Interpret your answer to (ii) in context.

   [6 marks]

   Total for question 4: 12 marks

End of questions