Graphs and transformations – Year 12 A level maths worksheet with answers to download as PDF

Sketch the curve \(y=(x+2)(x-1)(x-3)\). Label every point where it crosses the axes.

1. State the roots and the \(y\)-intercept. [3 marks]
2. Describe the end behaviour as \(x\to\pm\infty\). [2 marks]
3. Solve \((x+2)(x-1)(x-3)<0\) using your sketch. [2 marks]

Sketch the curve \(y=(x-2)^2(x+1)\).

1. Explain what the repeated factor \((x-2)^2\) tells you about the curve at \(x=2\). [2 marks]
2. Find the \(y\)-intercept. [1 mark]
3. For which values of \(x\) is \(y\ge0\)? [2 marks]
4. The curve is translated by \(\begin{pmatrix}0\\-4\end{pmatrix}\). Write down its new equation and find its \(y\)-intercept. [3 marks]

The curve \(C\) has equation \(y=\dfrac3{x-2}+1\).

1. Write down the equations of the two asymptotes. [2 marks]
2. Find the coordinates of the points where \(C\) meets the axes. [4 marks]
3. Describe the transformation that maps \(y=\dfrac1x\) to \(C\) in two steps. [3 marks]

The point \(A(4,6)\) lies on the curve \(y=f(x)\). Write down the coordinates of the image of \(A\) on each of these curves.

The curve \(C\) has equation \(y=x^2-3x\). Find the equation of the image of \(C\) after each transformation. Give each answer in the form \(y=ax^2+bx+c\).

1. a translation by \(\begin{pmatrix}2\\-1\end{pmatrix}\) [3 marks]
2. a reflection in the \(x\)-axis [1 mark]
3. a stretch with scale factor \(\tfrac12\) parallel to the \(x\)-axis [2 marks]
4. a reflection in the \(y\)-axis [2 marks]

Describe fully the single transformation that maps the graph of \(y=f(x)\) onto each of these graphs.

1. \(y=f(x+3)-4\) [2 marks]
2. \(y=3f(x)\) [2 marks]
3. \(y=f\!\left(\dfrac x2\right)\) [2 marks]
4. \(y=-f(x)\) [1 mark]
5. \(y=f(4x)\) [2 marks]

The main cable of a footbridge hangs between two towers 60 m apart. The towers are each 24 m tall (measured above the deck). The lowest point of the cable is 4 m above the deck, midway between the towers. Take the left tower at \(x=0\) and let \(y\) be the height above the deck.

1. The cable is a transformation of \(y=x^2\). Write the equation in the form \(y=a(x-p)^2+q\). [4 marks]
2. Describe the transformations from \(y=x^2\) in order. [3 marks]
3. Find the height of the cable 10 m from a tower. Give your answer to 2 decimal places. [2 marks]
4. A support rope is needed where the cable is 10 m above the deck. How far from a tower is it (2 decimal places)? [3 marks]

A taxi travels 120 km at an average speed of \(v\) km/h. The journey time is \(t=\dfrac{120}v\) hours. When there is a road closure the driver adds a 0.5 hour delay, so \(t=\dfrac{120}v+0.5\).

1. Find the journey time at 60 km/h and at 80 km/h without the delay. [2 marks]
2. Describe the transformation from \(t=\dfrac{120}v\) to \(t=\dfrac{120}v+0.5\). State the equations of the asymptotes of each graph. [3 marks]
3. With the delay, find the journey time at 100 km/h. [2 marks]
4. The passenger must arrive within 1.8 hours. Find the minimum average speed with the delay, to 1 decimal place. [3 marks]

A group of \(x\) friends share the cost of a £36 pizza order equally, so each pays \(y=\dfrac{36}x\) pounds. Delivery costs £3 per person, giving \(y=\dfrac{36}x+3\). A £6 discount code reduces the order to £30, giving \(y=\dfrac{30}x+3\).

1. Check the table entry for \(x=9\) and add the missing row for \(\dfrac{30}x+3\) when \(x=3,6,15\). [3 marks]
2. Describe how the graph of \(y=\dfrac{36}x+3\) is related to \(y=\dfrac1x\). [2 marks]
3. How many friends are needed for each to pay £5 without the code? [2 marks]
4. How many are needed for each to pay £5 with the code? [2 marks]

A cycle route follows the ground profile \(h=\dfrac16x(x-4)(x-9)\) where \(x\) is the horizontal distance in kilometres from the start and \(h\) is the height in tens of metres relative to sea level.

1. Show that the route is at sea level at \(x=0\), \(x=4\) and \(x=9\). [2 marks]
2. Sketch the profile for \(0\le x\le10\). [3 marks]
3. For which values of \(x\) in the range is the route above sea level? [2 marks]
4. Find the height at \(x=2\) in metres. [2 marks]

A fungus colony covers \(y=3^x\) square centimetres after \(x\) days. A second colony covers \(y=3^{x+1}\) square centimetres.

1. Sketch both curves on the same axes for \(-2\le x\le3\). Include the \(y\)-intercepts. [3 marks]
2. Show that \(3^{x+1}=3\times3^x\) and describe the transformation from the first to the second in two different ways. [3 marks]
3. State the equation of the asymptote. [1 mark]
4. After how many days is the second colony larger than 100 cm\(^2\)? Give your answer to 1 decimal place. [3 marks]

The curve \(y=x^2(x-3)\) models the water level in a tank over time. The table gives some values.

1. Check the values of \(y\) at \(x=-1\) and \(x=2\). [2 marks]
2. Sketch the curve, marking the turning points. [3 marks]
3. The line \(y=k\) is drawn. State the number of solutions of \(x^2(x-3)=k\) when \(k=-3\), \(k=-4\), \(k=0\) and \(k=5\). [4 marks]
4. Show that \(x^3-3x^2+4=0\) factorises as \((x+1)(x-2)^2\) and say what this means for \(k=-4\). [3 marks]

The temperature of a drink after \(t\) minutes is modelled by \(T=20+60\times2^{-t/5}\) degrees Celsius.

1. Verify the table at \(t=5\) and \(t=15\). [2 marks]
2. State the horizontal asymptote and explain its meaning in the context. [2 marks]
3. Describe the transformations that map \(y=2^t\) to \(T\) in order. [4 marks]
4. The drink is safe to hold when it is below 40°C. After how many minutes (to 1 decimal place)? [3 marks]

The diagram of \(y=f(x)\) (not shown to scale) has these key points: it crosses the \(x\)-axis at \(A(-3,0)\) and \(D(3,0)\), has a maximum at \(B(-1,4)\) and a minimum at \(C(1,-2)\).

1. For \(y=f(x)+3\) write down the coordinates of \(B\) and \(C\). [2 marks]
2. For \(y=f(x-2)\) write down the coordinates of all four points. [4 marks]
3. For \(y=2f(x)\) write down the new turning points. [2 marks]
4. For \(y=f(2x)\) write down the images of \(A\) and \(D\). [2 marks]
5. For \(y=-f(x)\) write down the new turning points and say which is a maximum. [3 marks]

\(f(x)=x^3-4x\).

1. Show that \(f(x)=x(x-2)(x+2)\) and sketch \(y=f(x)\). [3 marks]
2. On the same axes sketch \(y=f(x-1)\), marking the \(x\)-intercepts. [3 marks]
3. Find the \(y\)-intercept of \(y=f(x-1)\). [2 marks]
4. The curve \(y=f(x+2)\) passes through the origin. Write down its equation in factorised form and verify that it passes through \((0,0)\). [3 marks]

The curve \(C\) has equation \(y=\dfrac2{x-1}+3\) and the line \(L\) has equation \(y=x+1\).

1. Write down the equations of the asymptotes of \(C\). [2 marks]
2. Find where \(C\) crosses the axes. [3 marks]
3. Show that the \(x\)-coordinates of the points where \(L\) meets \(C\) satisfy \(x^2-3x=0\). [4 marks]
4. Find the coordinates of the points of intersection. [2 marks]

Find the equation of each curve described below.

1. A quadratic curve with minimum point \((3,-4)\) that passes through \((0,5)\). [4 marks]
2. A cubic curve that touches the \(x\)-axis at \(x=-2\), crosses it at \(x=1\) and passes through \((0,-8)\). [4 marks]
3. Find the height of the cubic in part (b) when \(x=2\). [1 mark]
4. The graph of part (a) is reflected in the \(x\)-axis. Write down the new equation and the coordinates of the maximum point. [2 marks]

An architect designs an arched window. The top edge of the glass is the curve \(y=9-x^2\) for \(-3\le x\le3\) (in feet), with the straight bottom edge on the \(x\)-axis.

1. Show that the curve is the image of \(y=x^2\) after a reflection in the \(x\)-axis followed by a translation. State the vector. [3 marks]
2. Find the height of the window at \(x=1\) and at \(x=2\). [2 marks]
3. The architect considers the wider arch \(y=9-\dfrac{x^2}4\). Describe the transformation from \(y=9-x^2\) to this curve and find the new width of the window at the bottom. [4 marks]
4. Both windows are 9 ft high in the middle. Find where the two curves meet and at what height. [4 marks]

\(f(x)=x^4-4x^2\).

1. Factorise \(f(x)\) completely and sketch the curve. [4 marks]
2. Show that \(f(-x)=f(x)\) and say what this means for the graph. [2 marks]
3. Solve \(f(x)<0\). [3 marks]
4. The curve is translated by \(\begin{pmatrix}0\\4\end{pmatrix}\). Show that the new equation can be written \(y=(x^2-2)^2\) and state the coordinates of the points where it touches the \(x\)-axis. [4 marks]

The curve \(y=f(x)\) is translated by \(\begin{pmatrix}2\\-1\end{pmatrix}\) to give the curve \(y=x^2-6x+12\).

1. Find \(f(x)\). [4 marks]
2. Show that neither curve crosses the \(x\)-axis. [3 marks]
3. The curve \(y=f(x)-k\) is a translation of \(y=f(x)\) vertically. Find the set of values of \(k\) for which it meets the \(x\)-axis at two distinct points. [4 marks]
4. Find the translation vector that moves \(y=f(x)\) so that its turning point lies on the \(x\)-axis. [2 marks]