Combining graph transformationsEdexcel International A Level Maths: Subtopic test

1. 1
   The curve $y=f(x)$ has a maximum point at $(2,5)$ and crosses the $y$-axis at $(0,1)$.

   (a)

   Which are the coordinates of the maximum point on the curve $y=2f(3x)$?

   [1 mark]

   • A$(6,10)$
   • B$\left(\frac23,10\right)$
   • C$\left(\frac23,5\right)$
   • D$(6,5)$

   (b)

   Which are the coordinates of the maximum point on the curve $y=f(-x)+1$?

   [1 mark]

   • A$(2,6)$
   • B$(-2,5)$
   • C$(-2,4)$
   • D$(-2,6)$

   (c)

   Find the coordinates of the turning point of the curve $y=3-f\left(\frac{x}{2}\right)$ and state whether it is a maximum or minimum.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The point $A(4,-6)$ lies on the curve $y=f(x)$.

   (a)

   Which are the coordinates of the image of $A$ on the curve $y=f(x-3)+2$?

   [1 mark]

   • A$(7,-4)$
   • B$(1,-4)$
   • C$(7,-8)$
   • D$(1,-8)$

   (b)

   Which are the coordinates of the image of $A$ on the curve $y=-2f(x)$?

   [1 mark]

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

   (c)

   Write down the coordinates of the image of $A$ on the curve $y=-f(-x)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $g$ is defined by $g(x)=3+\sin2x$ for $0\leqslant x\leqslant2\pi$, where $x$ is in radians. It is formed by transforming $y=\sin x$.

   (a)

   Describe the sequence of transformations that maps $y=\sin x$ to $y=g(x)$, and state the period of $g$.

   [3 marks]

   (b)

   Find the maximum and minimum values of $g(x)$, and the values of $x$ in the interval $0\leqslant x\leqslant2\pi$ at which they occur.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $f(x)=x^2-2x-3$, $x\in\mathbb{R}$.

   (a)

   (i) Write down the coordinates of the minimum point of $y=f(x)$.
   (ii) The curve $y=f(x)$ is transformed to $y=2f(x+1)$. Find the coordinates of the turning point and of the $x$-intercepts of the new curve.
   (iii) Find the equation of the new curve in the form $y=ax^2+b$.

   [6 marks]

   (b)

   The curve $y=f(x)$ is reflected in the $x$-axis and then translated by $\begin{pmatrix} 0 \\ 5 \end{pmatrix}$.
   (i) Find the equation of the resulting curve.
   (ii) Find the coordinates of its maximum point.
   (iii) Find the $x$-intercepts of the resulting curve.

   [6 marks]

   Total for question 4: 12 marks

End of questions