Graphs and transformationsAQA A-Level Maths: Topic test

1. 1
   The function $g$ is defined by $g(x)=x^3-4x$ for $x\in\mathbb{R}$.

   (a)

   How many times does the curve $y=g(x)$ cross the $x$-axis?

   [1 mark]

   • A1
   • B2
   • C3
   • D4

   (b)

   The curve $y=g(x)$ is translated by 2 units in the positive $x$-direction. Which is the equation of the new curve?

   [1 mark]

   • A$y=(x-2)^3-4(x-2)$
   • B$y=(x+2)^3-4(x+2)$
   • C$y=x^3-4x-2$
   • D$y=x^3-4x+2$

   (c)

   Find the $x$-coordinates of the points where the curve $y=g(x)$ crosses the $x$-axis, and state what happens to $y$ as $x\to-\infty$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $K$ has equation $y=\dfrac{5}{x^2}-2$ for $x\neq0$.

   (a)

   Which line is a horizontal asymptote of $K$?

   [1 mark]

   • A$x=0$
   • B$y=2$
   • C$y=5$
   • D$y=-2$

   (b)

   $K$ is obtained from the curve $y=\dfrac{1}{x^2}$ by two transformations. Which pair, in the order given, produces $K$?

   [1 mark]

   • Aa stretch with scale factor 5 parallel to the $y$-axis, then a translation of 2 units in the negative $y$-direction
   • Ba stretch with scale factor 5 parallel to the $x$-axis, then a translation of 2 units in the negative $y$-direction
   • Ca translation of 2 units in the negative $y$-direction, then a stretch with scale factor 5 parallel to the $y$-axis
   • Da stretch with scale factor 5 parallel to the $y$-axis, then a translation of 2 units in the negative $x$-direction

   (c)

   Find the $x$-coordinates of the points where $K$ crosses the $x$-axis.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=2x^2-3x-5$ and the line $l$ has equation $y=x+3$.

   (a)

   Find the exact $x$-coordinates of the points where $C$ and $l$ intersect.

   [3 marks]

   (b)

   The line $y=x+k$ is a tangent to $C$. Find the value of $k$ and the coordinates of the point of contact.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The function $f$ is defined by $f(x)=x^2-6x+10$ for $x\in\mathbb{R}$. A designer models the height of a hanging cable above level ground, in metres, at a horizontal distance $x$ metres from one support, by $y=f(x)$ for $0\le x\le6$.

   (a)

   The curve $y=f(x)$ is stretched with scale factor 2 parallel to the $y$-axis and then translated by 1 unit in the negative $x$-direction and 4 units in the positive $y$-direction. The new curve is $y=g(x)$. Find the coordinates of the minimum point of $y=g(x)$ and show that $g(x)=2x^2-8x+14$.

   [6 marks]

   (b)

   The cable is attached to supports at $x=0$ and $x=6$. A beam is fixed 5 m above the ground. Using the model, find the horizontal distance between the two points where the cable is 5 m above the ground. A measurement shows that the cable is actually 8 m above the ground at $x=0$. Show that the model overestimates this height by 2 m, then find the value of $a$ in the refined model $y=a(x-3)^2+1$ that gives a height of 8 m at $x=0$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The curve $P$ has equation $y=x^4-4x^2$ for $x\in\mathbb{R}$.

   (a)

   Which statement about $P$ at the origin is correct?

   [1 mark]

   • A$P$ crosses the $x$-axis at the origin.
   • B$P$ touches the $x$-axis at the origin, where it has a turning point.
   • C$P$ has a vertical asymptote at the origin.
   • D$P$ does not pass through the origin.

   (b)

   $P$ is stretched with scale factor $\dfrac{1}{2}$ parallel to the $x$-axis. Which is the equation of the new curve?

   [1 mark]

   • A$y=\dfrac{x^4}{16}-x^2$
   • B$y=16x^4-16x^2$
   • C$y=2x^4-8x^2$
   • D$y=\dfrac{x^4}{2}-2x^2$

   (c)

   Find the $x$-coordinates of the points where $P$ meets the line $y=-4$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The curve $y=f(x)$ passes through the points $(-4,3)$, $(0,0)$ and $(2,-5)$.

   (a)

   Which point lies on the curve $y=f(x+3)$?

   [1 mark]

   • A$(-1,3)$
   • B$(-4,6)$
   • C$(-7,3)$
   • D$(-4,0)$

   (b)

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

   [1 mark]

   • A$(4,-5)$
   • B$(1,-5)$
   • C$(2,-7)$
   • D$(2,-10)$

   (c)

   Write down the coordinates of the image of the point $(2,-5)$ on the curve $y=3f(x-1)+2$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The weekly profit, $P$ thousand pounds, of a seasonal product is modelled by $P=-2t^2+16t-24$, where $t$ is the number of weeks after launch and $0\le t\le8$.

   (a)

   Find the greatest weekly profit predicted by the model and the value of $t$ at which it occurs.

   [3 marks]

   (b)

   Find the values of $t$ for which the model predicts a profit, and comment on what the model predicts at $t=0$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The curve $C$ has equation $y=x^2-4x+3$. The curve $D$ has equation $y=\dfrac{k}{x}$ for $x\neq0$, where $k$ is a constant.

   (a)

   The curve $C$ is translated by 2 units in the positive $x$-direction and then stretched with scale factor 3 parallel to the $y$-axis. The resulting curve is $E$. Find the equation of $E$ and the $x$-coordinates of the points where $E$ crosses the $x$-axis.

   [6 marks]

   (b)

   The curve $D$ passes through the minimum point of $C$. Find the value of $k$, and hence find the $x$-coordinates of all the points where $C$ and $D$ intersect.

   [6 marks]

   Total for question 8: 12 marks

End of questions