Find the equation of each graph after the given transformation.

(d) $y=2x^2-8x+10$ after a stretch parallel to the $x$ -axis with stretch factor $2$.

The diagram shows the graph of $y=g\left(x\right)$. Sketch the graph of each of the following.

 $y=\mathrm{g}(x+2)+3$.

Find the equation of the image of the curve $y=x^2$ after each of the following combinations of transformations and, in each case, sketch the graph of the resulting curve.

(a) A stretch in the $x$ -direction with factor 2 followed by a translation by the vector $\left(\begin{array}{l}5\\ 0\end{array}\right)$.

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined for $x\in \mathrm{ℝ}$ by:

$\mathrm{f}:x\mapsto 3x-1$

$\mathrm{g}:x\mapsto 5x-x^2$

Express $\mathrm{g}f\left(x\right)$ in the form $a-b(x-c{)}^2$, where $a,b \& c$ are constants.

The diagram shows a sketch of the curve with equation $y=\mathrm{f}\left(x\right)$.

(b) Describe fully a sequence of two transformations that maps the graph of $y=f\left(x\right)$ onto the graph of $y=f(3-x)$

A curve has equation $y=x^2+6x+8$.

Sketch the curve, showing the coordinates of any axes crossing points.

The function $f:x\mapsto x^2-2$ is defined for the domain $x⩾0$.

(a) Find $f^{-1}\left(x\right)$ and state the domain of $f^{-1}$.

The function $f:x\mapsto x^2-2$ is defined for the domain $x⩾0$.

On the same diagram, sketch the graphs of $\mathrm{f}$(x) and ${\mathrm{f}}^{-1}$(x).