Quick questions on Graphs and transformations: curve sketching, translations, stretches and reflections - OCR A-Level Maths A

For a polynomial, find where it crosses the axes and the general shape from the leading term. A cubic with positive leading coefficient runs from bottom-left to top-right; repeated roots touch the axis rather than crossing it. Mark the $y$-intercept (set $x = 0$) and the roots (set $y = 0$).

The "inside the bracket" transformations ($f(x + a)$, $f(ax)$) act on $x$ and behave oppositely to intuition: $f(x + a)$ moves left, and $f(ax)$ compresses by factor $1/a$. The "outside" transformations ($f(x) + a$, $af(x)$) act on $y$ as expected. A negative scale factor reflects: $-f(x)$ reflects in the $x$-axis and $f(-x)$ reflects in the $y$-axis.

Many OCR questions give only a sketch of $y = f(x)$ with named features (turning points, intercepts, asymptotes) and ask for the same features on a transformed graph. The reliable method is to apply the transformation rule to each named coordinate in turn, remembering that vertical stretches fix points on the $x$-axis (since the $y$-coordinate is zero) and horizontal stretches fix points on the $y$-axis. Asymptotes transform like the curve: a vertical asymptote shifts under a horizontal translation, and a horizontal asymptote shifts under a vertical translation.

Describe the transformation taking $y = f(x)$ to $y = f(x) - 4$. [1 mark]

The graph $y = \sin x$ is stretched to give $y = \sin 2x$. State the new period. [2 marks]