![graph of y = f of x](/combining-functions/r9673/images/img-9673-1.png)
The diagram shows the graph for \(y = f(x)\). The curve passes through the origin, and has a maximum point at \((1, 1)\). Sketch, on separate diagrams, the graphs of
- \(y = f(x) + 2\),
… giving the coordinates of the maximum point in each case.
The graph of the function \(y = f(x) + 2\) can be obtained from the graph of the original function \(f\) by raising the \(y\)-coordinate of each point on the graph by \(2\). In particular, the maximum becomes \((1, 3)\). The new graph looks like this.
![graph of y = f(x) + 2](/combining-functions/r9673/images/img-9673-2.png)
- \(y = f(x + 2)\),
You might expect a shift to the right, but if you think that the \(y\)-coordinate associated with \(x = 0\) on the old graph is now associated with \(x = -2\) on the new graph, you can see it is a translation to the left.
In particular, the maximum becomes \((-1, 1)\). The new graph looks like this.
![graph of y = f(x + 2)](/combining-functions/r9673/images/img-9673-3.png)
- \(y = f(2x)\).
The graph of the function \(y = f(2x)\) can be obtained from the graph of the original function \(f\) by compressing the graph by a scale factor of \(2\) in the \(x\)-direction.
We could describe this as a stretch in the \(x\)-direction with scale factor \(\dfrac{1}{2}\).
In particular, the maximum becomes \((0.5, 1)\). The new graph looks like this.
![graph of y = f(2x)](/combining-functions/r9673/images/img-9673-4.png)
Can you think of an \(f(x)\) that would give something like the starting graph above? You might use graphing software to try a few functions.