1. Here is part of a graph of $y = \mathrm{f}(x)$. Sketch the graph of $y = \mathrm{f}(x) + 5$, labelling any intersections with the coordinate axes.
2. Here is part of a graph of $y = \mathrm{f}(x)$. Sketch the graph of $y = \mathrm{f}(x-2)$, showing clearly where the labelled points end up.
3. Here is part of a graph of $y = \mathrm{f}(x)$. Sketch the graph of $y = 2\mathrm{f}(x)$, showing clearly where the labelled points end up.
4. Here is part of a graph of $y = \mathrm{f}(x)$. Sketch the graph of $y = \mathrm{f}(0.5x)$, showing clearly where the labelled points end up.
5. Describe the transformation that maps the solid line onto the dashed line.
In any order Stretch parallel to $x$ axis, scale factor $0.5$ Stretch parallel to $y$ axis, scale factor $2$
6. Describe the transformation that maps the solid line onto the dashed line.
In any order Stretch parallel to $x$ axis, scale factor $2$ Translation down by $2$ units
7. Sketch, on the same set of axes, the following in the range $-\pi < x \leqslant \pi$ $$y = \cos x$$ $$y = 2\cos(x+\pi)$$
8. Here is part of a graph of $y = \mathrm{f}(x)$. Sketch the graph of $y = \mathrm{f}(-0.5x)+1$, showing clearly where the labelled points end up.
9. A curve $y = \mathrm{f}(x)$ has a turning point at $(1,-9)$. Write down the location of the turning point of $$y = -2\mathrm{f}(x+3)+1$$
$(-2,19)$
10. A curve $y = \mathrm{f}(x)$ has a turning point at $(-1,16)$. Write down the location of the turning point of $$y = 0.5\mathrm{f}(1-x)$$
$(2,8)$
11. The point $(1,-32)$ lies on the curve $y = \mathrm{f}(x)$. Write down the corresponding point on the curve $$y = 3-\mathrm{f}(2-2x)$$
$(0.5,35)$
12. The point $(1,1)$ lies on the curve $y = \mathrm{f}(x)$. Write down the corresponding point on the curve $$y = \mathrm{f}(2(x-3))+1$$
$(3.5,2)$
13. The following three transformations are applied to a curve $y = \mathrm{f}(x)$ in order. Write down the equation of the new curve. Translation by $\begin{pmatrix}1\\-4\end{pmatrix}$ Stretch parallel to $x$ axis, scale factor $2$ Translation by $\begin{pmatrix}2\\0\end{pmatrix}$
$\mathrm{f}(0.5x-2)-4$
14. The following three transformations are applied to a curve $y = \mathrm{f}(x)$ in order. Write down the equation of the new curve. Stretch parallel to $x$ axis, scale factor $0.25$ Translation by $\begin{pmatrix}2\\0\end{pmatrix}$ Stretch parallel to $x$ axis, scale factor $2$
$\mathrm{f}(2x-8)$
15. Given $$\mathrm{f}(x) = 3x^2 + 3x - 1$$ write down the equation of $2\mathrm{f}(2x)+3$ in the form $ax^2 + bx + c$.
$24x^2+12x+1$
16. Given $$\mathrm{f}(x) = 5-2x^2 + x$$ write down the equation of $\mathrm{f}(3x-2)+1$ in the form $ax^2 + bx + c$.
$-18x^2+27x-4$
17. A graph of $$y = 3x^2 + 4x$$ is translated by the vector $\begin{pmatrix}2\\1\end{pmatrix}$, then stretched parallel to the $x$ axis with scale factor $3$. Write down the new equation of the curve.
$y = \dfrac{1}{3}x^2 - \dfrac{8}{3}x+5$
18. A graph of $$y = x^2 + 7x - 2$$ is stretched parallel to the $x$ axis with scale factor $\dfrac{1}{4}$, then translated by the vector $\begin{pmatrix}-1\\-3\end{pmatrix}$. Write down the new equation of the curve.
$y = 16x^2 + 60x + 39$
19. The image of the curve $$y = x^2 + 8x - 4$$ after a transformation has the equation $$y = x^2 - 6x - 4$$ Describe the transformation
Translation by the vector $\begin{pmatrix}7\\7\end{pmatrix}$
20. The curve $$y = x^2 - 4x$$ is mapped onto the curve $$y = 2x^2 + 8x - 3$$ first by a stretch parallel to the $y$ axis, then by a translation. Describe the translation.
$\begin{pmatrix}-4\\-3\end{pmatrix}$
21. A curve $y = x^3$ is translated to the right by $1$ unit, stretched parallel to the $x$ axis with scale factor $2$, then translated to the left by $2$ units. Write down a single transformation in the $y$ direction that has the same result.
Stretch parallel to the $y$ axis with scale factor $\frac{1}{8}$ or Stretch parallel to the $x$ axis with scale factor $2$
22. Describe the transformation that maps the curve $$y = \dfrac{1}{x^2}$$ onto the curve $$y = \dfrac{1}{x^2 + 6x + 9}$$
Translation $\begin{pmatrix}-3\\0\end{pmatrix}$
23. Describe a sequence of individual transformations that would map a graph of $$y = \mathrm{f}(2x) + 1$$ onto a graph of $$y = \mathrm{f}(0.5x + 1)$$
Translation $\begin{pmatrix}-0.5\\-1\end{pmatrix}$ Stretch parallel to $x$ axis with scale factor $4$
24. Describe a sequence of individual transformations that would map a graph of $$y = \mathrm{f}(4x) + 1$$ onto a graph of $$y = 3\mathrm{f}(2x - 3)$$
Translation $\begin{pmatrix}\frac{3}{4}\\-1\end{pmatrix}$ Stretch parallel to $x$ axis with scale factor $2$ Stretch parallel to $y$ axis with scale factor $3$
