Composite transformations Problem set A

Write down the equation of the graph after each change.

1. (a) The graph of \(y=5x^4\) is stretched vertically with scale factor 6.
2. (b) The graph of \(y=4 x^{3} - x^{2}\) is reflected in the x-axis and then reflected in the y-axis.
3. (c) The graph of \(y=3^x\) is translated 2 units in the positive x direction and then stretched vertically with scale factor 5.
4. (d) The graph of \(y=x^{2} - 4 x\) is translated 1 unit in the negative x direction and then stretched vertically with scale factor 2. Give your answer in the form \(y=ax^2+bx+c\).

A vertical stretch can sometimes be replaced by a horizontal stretch.

1. (a) The graph of \(y=x^2\) is stretched vertically with scale factor 16. Write down the equation of the new graph.
2. (b) Find the scale factor of the horizontal stretch that has the same effect on \(y=x^2\).
3. (c) The graph of \(y=2x^3\) is stretched vertically with scale factor 125. Find the scale factor of the horizontal stretch that has the same effect.

The diagram shows the graph of \(y=f(x)\), a cubic. On separate diagrams sketch each graph, marking the turning points.

1. (a) \(y=3f(x+1)\)
2. (b) \(y=\tfrac12f(x-4)\)

The diagram shows the graph of \(y=f(x)\), a cubic. On separate diagrams sketch each graph, marking the turning points.

1. (a) \(y=-f(x)+2\)
2. (b) \(y=1-2f(x)\)

The diagram shows the graph of \(y=f(x)\), a cubic. On separate diagrams sketch each graph, marking the turning points.

1. (a) \(y=f(\tfrac12x-1)\)
2. (b) \(y=f(2x-3)\)

The graph of \(y=f(x)\) is made of straight lines joining (−4, 1), (−2, −3), (0, −3), (2, 2), (4, 2). Sketch each graph.

1. (a) \(y=2f(x-1)+1\)
2. (b) \(y=f(2x+4)\)

Describe a sequence of transformations that maps the graph of \(y=2 x^{2} - x\) to the graph of the given equation.

1. (a) \(y=6 x^{2} - 3 x + 4\)
2. (b) \(y=2 x^{2} - 17 x + 37\)

The graph of \(f(x)=3^x\) has the horizontal asymptote \(y=0\). For each change write the equation of the image and the equation of its horizontal asymptote.

1. (a) A translation 2 units up.
2. (b) A reflection in the x-axis, then a translation 5 units up.
3. (c) A vertical stretch with scale factor 4, then a translation 1 unit down.

The graph of \(y=\dfrac{1}{x}\) has the asymptotes \(x=0\) and \(y=0\). The graph is translated 3 units to the left and 2 units down. Write the equation of the image and the equations of its asymptotes.

The curve of a suspended cable between two towers is modelled by \(y=\frac{x^{2}}{2} - 6 x + 21\), where x and y are in metres.

1. (a) Write \(\frac{x^{2}}{2} - 6 x + 21\) in the form \(a(x-h)^2+k\).
2. (b) Describe two transformations that map the graph of \(y=x^2\) to this curve.

A graph shows the temperature C in °C of a room against time. The temperature in °F is \(F=1.8C+32\).

1. (a) Describe the transformations that map the graph of C to the graph of F.
2. (b) The same graph can be made by translating first and stretching second. Describe that translation and that stretch.

Describe the transformations that map \(y=\sqrt{x}\) to each graph, in order.

1. (a) \(y=\sqrt{5x-3}\)
2. (b) \(y=\sqrt{5(x-3)}\)
3. (c) Where does the point \((0,0)\) of \(y=\sqrt{x}\) end up in each graph?

Find two different sequences of two transformations that map \(y=f(x)\) to \(y=f(6x+3)\).

The graph of \(y=ax^2+bx+c\) is translated 1 unit in the negative x direction and then stretched horizontally with scale factor \(\tfrac12\). The resulting graph is \(y=4x^2+cx+17\). Find a, b and c.

The graph of \(y=f(x)\) is stretched vertically with scale factor 5 away from the line \(y=-1\) (the line \(y=-1\) does not move).

1. (a) Write down the equation of the new graph in terms of f.
2. (b) The point (3, 4) is on the graph of \(y=f(x)\). Find its image.