Functions and graphsIB MYP Maths Standard: Topic test

1. 1
   A function is defined by $g(x)=2x^{2}-5$.

   (a)

   What is the value of $g(-3)$?

   [1 mark]

   • A$13$
   • B$31$
   • C$-23$
   • D$-11$

   (b)

   Which values of $x$ satisfy $g(x)=45$?

   [1 mark]

   • A$x=\pm\sqrt{22.5}$
   • B$x=5$ only
   • C$x=25$
   • D$x=5$ and $x=-5$

   (c)

   Show that $g(4)=g(-4)$, and explain why this does not stop $g$ from being a function.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The line $L$ passes through the points $(-1,\ 7)$ and $(3,\ -1)$.

   (a)

   What is the gradient of $L$?

   [1 mark]

   • A$2$
   • B$-\frac{1}{2}$
   • C$-2$
   • D$\frac{1}{2}$

   (b)

   Which is the equation of the line parallel to $L$ that passes through $(0,\ 4)$?

   [1 mark]

   • A$y=2x+4$
   • B$y=-2x+4$
   • C$y=-\frac{1}{2}x+4$
   • D$y=-2x+5$

   (c)

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

   [2 marks]

   Total for question 2: 4 marks

3. 3
   An electric car in Norway is plugged in to charge. The battery charge $C$ (as a percentage of a full battery) after $t$ hours of charging is modelled by $C=20+12.5t$, until the battery is full.

   (a)

   (i) Explain what the numbers $20$ and $12.5$ mean in this context. (ii) Find the charge after $3$ hours.

   [3 marks]

   (b)

   Find how long the battery takes to be fully charged, in hours and minutes. State the values of $t$ for which the model is valid, and give a reason.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   On a map of a nature reserve, one unit on each axis is $1$ km. A ranger station is at $R(1,\ 2)$ and a lookout tower is at $T(9,\ 8)$. A well is at $W(8,\ 1)$.

   (a)

   (i) Find the coordinates of the midpoint of $RT$. (ii) Find the distance from the ranger station to the lookout tower. (iii) Find the gradient of $RT$.

   [6 marks]

   (b)

   A new path starts at the midpoint $M(5,\ 5)$ of $RT$ and is perpendicular to $RT$. (i) Find the equation of the path in the form $y=mx+c$. (ii) Show that the well $W$ is on the path. (iii) Show that $W$ is the same distance from $R$ as from $T$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A quadratic function is $y=x^{2}-6x+5$.

   (a)

   At which values of $x$ does the graph cross the $x$-axis?

   [1 mark]

   • A$x=-1$ and $x=-5$
   • B$x=1$ and $x=5$
   • C$x=1$ and $x=-5$
   • D$x=6$ and $x=5$

   (b)

   What is the equation of the axis of symmetry of the graph?

   [1 mark]

   • A$x=3$
   • B$x=5$
   • C$x=-3$
   • D$x=6$

   (c)

   Find the coordinates of the vertex of the graph.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A student uses graphing software to plot $y=x^{2}-4$ and $y=x+2$ on the same axes.

   (a)

   At which values of $x$ do the two graphs intersect?

   [1 mark]

   • A$x=3$ only
   • B$x=2$ and $x=-3$
   • C$x=-2$ and $x=3$
   • D$x=-4$ and $x=2$

   (b)

   The student then plots a third line, $y=x-8$. How many points does it share with the parabola $y=x^{2}-4$?

   [1 mark]

   • AOne
   • BTwo
   • CInfinitely many
   • DNone

   (c)

   Write down the coordinates of both points where $y=x^{2}-4$ and $y=x+2$ intersect.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   An algorithm is written as follows. Step 1: Input a positive whole number $n$. Step 2: Set $s=0$. Step 3: Add $n$ to $s$. Step 4: Subtract $2$ from $n$. Step 5: If $n>0$, go back to Step 3; otherwise output $s$.

   (a)

   Trace the algorithm for the input $n=7$, showing the value of $s$ each time Step 3 is carried out, and state the output.

   [3 marks]

   (b)

   Describe what the algorithm calculates when the input $n$ is an odd number. Use your description to find the output when $n=15$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Two towns, Qasr and Ramla, are $120$ km apart on a straight road. A bus leaves Qasr at 08:00 and travels towards Ramla at a constant speed of $60$ km/h. A car leaves Ramla at 08:30 and travels towards Qasr at a constant speed of $90$ km/h. Let $t$ be the number of hours after 08:00 and let $d$ be the distance in km from Qasr.

   (a)

   (i) Write an equation for $d$ in terms of $t$ for the bus, and another for the car, once both are travelling. (ii) Solve the equations to find when the bus and the car meet. (iii) State the time they meet and their distance from Qasr.

   [6 marks]

   (b)

   (i) Explain what the gradient $-90$ in the equation of the car means. (ii) Find the time at which the car reaches Qasr and the time at which the bus reaches Ramla. (iii) Which vehicle completes its journey first, and by how many minutes?

   [6 marks]

   Total for question 8: 12 marks

End of questions