The graph of $y = f(x)$ is shown below - AQA - A-Level Maths Mechanics - Question 6 - 2020 - Paper 3

To sketch the graph of $y = 2f(x) - 4$, we firstly stretch the graph of $f(x)$ vertically by a factor of $2$, which multiplies all $y$ values by $2$. Then, we translate the entire graph downwards by $4$ units.

For example:

• The point $(0, 2)$ becomes $(0, 2 imes 2 - 4) = (0, 0)$.
• The point $(2, 6)$ transforms to $(2, 2 imes 6 - 4) = (2, 8)$.
• The point $(-1, 0)$ becomes $(-1, 2 imes 0 - 4) = (-1, -4)$.
• The point $(-2, -2)$ transforms to $(-2, 2 imes -2 - 4) = (-2, -8)$.

Finally, connect these points smoothly to draw the transformed graph.

To sketch the graph of $y = f'(x)$, we need to determine the slope (derivative) of the function $f(x)$ at various points. We analyze the segments of the original graph and find the following:

• Between $x = -2$ and $x = -1$, the graph is horizontal, so the derivative is $0$.
• For $-1 < x < 0$, the line is increasing with a positive slope, hence $f'(x) > 0$.
• From $x = 0$ to $x = 2$, the graph again shows an increasing linear trend, providing a positive derivative.
• Beyond $x = 2$, the graph becomes horizontal, giving $f'(x) = 0$.

We can mark the critical points on the $x$-axis and horizontal lines where necessary, indicating the behavior of the derivative at different intervals.