Transformations of graphsAQA A-Level Maths: Subtopic test

1. 1
   The curve $y=f(x)$ has a minimum point at $(3,-2)$.

   (a)

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

   [1 mark]

   • A$(3,-6)$
   • B$(7,-2)$
   • C$(-1,-2)$
   • D$(3,2)$

   (b)

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

   [1 mark]

   • A$(7,-2)$
   • B$(-1,-2)$
   • C$(3,2)$
   • D$\left(\frac34,-2\right)$

   (c)

   Write down the coordinates of the minimum point of the curve $y=2f(x-1)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $y=f(x)$ crosses the $x$-axis at $(-2,0)$ and $(3,0)$ only, and crosses the $y$-axis at $(0,6)$.

   (a)

   Find the $x$-intercepts of the curve $y=f(2x)$.

   [1 mark]

   • A$x=-4$ and $x=6$
   • B$x=-2$ and $x=3$
   • C$x=-1$ and $x=\frac32$
   • D$x=-1$ and $x=3$

   (b)

   Find the $y$-intercept of the curve $y=-f(x)$.

   [1 mark]

   • A$(0,-6)$
   • B$(0,6)$
   • C$(6,0)$
   • D$(-6,0)$

   (c)

   Describe fully the single transformation that maps $y=f(x)$ onto $y=f(x+2)$, and write down the $x$-intercepts of $y=f(x+2)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $f$ is defined by $f(x)=x^2-4x+7$.

   (a)

   Show that the curve $y=f(x+1)$ has equation $y=x^2-2x+4$.

   [3 marks]

   (b)

   The curve $y=f(x)$ is transformed to $y=f(2x)$. Find the coordinates of the minimum point and of the $y$-intercept of $y=f(2x)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $y=f(x)$, where $f(x)=3-2x-x^2$, has a maximum point at $(-1,4)$ and crosses the $x$-axis at $(-3,0)$ and $(1,0)$.

   (a)

   For each curve, write down the coordinates of the maximum point and of the $x$-intercepts: (i) $y=2f(x)$, (ii) $y=f(x-2)$, (iii) $y=f(3x)$.

   [6 marks]

   (b)

   (i) Find the equation of the curve $y=f(x-2)$ in the form $y=-x^2+px+q$.
   (ii) Write down the $y$-intercept of $y=f(x-2)$.
   (iii) The curve $y=f(x)+k$ touches the $x$-axis. Find $k$ and describe the transformation.

   [6 marks]

   Total for question 4: 12 marks

End of questions