Combinations of transformationsAQA A-Level Maths: Subtopic test

1. 1
   The curve $y=f(x)$ has a minimum point at $(1,-3)$.

   (a)

   Find the coordinates of the minimum point of the curve $y=2f(x)+1$.

   [1 mark]

   • A$(1,-5)$
   • B$(1,-7)$
   • C$(1,-2)$
   • D$(2,-5)$

   (b)

   Find the coordinates of the minimum point of the curve $y=f(2x-4)$.

   [1 mark]

   • A$(6,-3)$
   • B$\left(-\frac32,-3\right)$
   • C$\left(\frac52,-3\right)$
   • D$\left(\frac12,-3\right)$

   (c)

   Write down the coordinates of the turning point of the curve $y=3-f(x)$ and state whether it is a maximum or a minimum.

   [2 marks]

   Total for question 1: 4 marks

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

   (a)

   Find the coordinates of the maximum point of the curve $y=g(x+3)-2$.

   [1 mark]

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

   (b)

   Which statement about the curve $y=-g(2x)$ is correct?

   [1 mark]

   • AIt has a minimum at $(-4,-5)$
   • BIt has a maximum at $(-1,-5)$
   • CIt has a minimum at $(-1,5)$
   • DIt has a minimum at $(-1,-5)$

   (c)

   Write down the coordinates of the maximum point of the curve $y=2g(x-1)-3$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=x^2$. The curve $D$ has equation $y=2(x-3)^2+1$.

   (a)

   Describe a sequence of three transformations that maps $C$ onto $D$.

   [3 marks]

   (b)

   Find the coordinates of the point where $D$ meets the $y$-axis. Find the coordinates of the minimum point of $D$ and hence explain why $D$ does not meet the $x$-axis.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $y=f(x)$ has a maximum point at $(2,6)$, crosses the $x$-axis at $(-1,0)$ and $(5,0)$, and crosses the $y$-axis at $(0,4)$.

   (a)

   Write down the coordinates of the stated points on each curve: (i) $y=3f(x-1)$: the maximum point and the $x$-intercepts; (ii) $y=-f\left(\frac x2\right)$: the minimum point and the $x$-intercepts; (iii) $y=f(2x)+1$: the maximum point and the $y$-intercept.

   [6 marks]

   (b)

   The curve $y=f(x)$ is transformed to the curve $y=af(x+b)+c$. The maximum point $(2,6)$ is mapped to $(-1,10)$ and the point $(5,0)$ is mapped to $(2,-2)$. Find the values of $a$, $b$ and $c$.

   [6 marks]

   Total for question 4: 12 marks

End of questions