Transformations of functionsIB MYP Maths Extended: Subtopic test

1. 1
   The graph of $y=f(x)$ has a minimum point at $(2,-1)$.

   (a)

   Where is the minimum point of the graph of $y=f(x)+3$?

   [1 mark]

   • A$(5,-1)$
   • B$(2,-4)$
   • C$(-1,-1)$
   • D$(2,2)$

   (b)

   Where is the minimum point of the graph of $y=f(x-4)$?

   [1 mark]

   • A$(6,-1)$
   • B$(-2,-1)$
   • C$(2,-5)$
   • D$(2,3)$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The point $(4,6)$ lies on the graph of $y=f(x)$.

   (a)

   Which point lies on the graph of $y=2f(x)$?

   [1 mark]

   • A$(8,6)$
   • B$(4,12)$
   • C$(4,3)$
   • D$(4,8)$

   (b)

   Which point lies on the graph of $y=f(2x)$?

   [1 mark]

   • A$(8,6)$
   • B$(4,3)$
   • C$(2,6)$
   • D$(-4,6)$

   (c)

   Find the coordinates of the image of $(4,6)$ on the graph of $y=-f(x+1)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The graph of $y=x^2$ is transformed to give the graph of $y=(x+3)^2-5$.

   (a)

   Describe the single transformation, and write down the coordinates of the vertex of the new graph.

   [3 marks]

   (b)

   Find the exact coordinates of the points where the new graph crosses the axes.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A designer models the profile of a bridge arch by $y=f(x)$, where $f(x)=9-x^2$ for $-3\leq x\leq3$. Both $x$ and $y$ are in metres, and the arch rests on the ground at $y=0$.

   (a)

   (i) The designer considers the arch $y=f\left(\frac{x}{2}\right)$. Describe the transformation and find the width of this arch at ground level.
   (ii) She also considers $y=1.5f(x)$. Write down its equation and find the height of this arch.

   [6 marks]

   (b)

   The arch is instead built on piers, so that its edges are $4$ m above the river, using the curve $y=f(x)+4$.
   (i) State the transformation, and find the height of the arch above the river.
   (ii) Its reflection in the calm water (the line $y=0$) is modelled by reflecting the arch in the $x$-axis. Find the equation of the reflection and its lowest point.
   (iii) Evaluate how well the model fits the piers, and give one limitation of the model.

   [6 marks]

   Total for question 4: 12 marks

End of questions