1. The graph \(y = f(x)\) of a certain function has been plotted below.

    Graph of f, a cubic S-shaped curve starting in the third quadrant and ending in the first quadrant, intersecting the x-axis at a negative value and touching the x-axis at a positive value

    On the next three pairs of axes (A), (B), (C) are graphs of \[y = f(-x), \quad f(x-1), \quad -f(x)\] in some order. Say which axes correspond to which graphs.

  2. Sketch graphs of both of the following functions \[y = 2^{-x^2} \quad \text{and} \quad y = 2^{2x - x^2}.\] Carefully label any stationary points.

  3. Let \(c\) be a real number and define the following integral \[I(c) = \int^1_0 2^{-(x-c)^2} dx.\]

    State the value(s) of \(c\) for which \(I(c)\) is largest. Briefly explain your reasoning. [Note you are not being asked to calculate this maximum value.]