Sketching curves including polynomials, the reciprocal function and its variations, intersections of graphs, and the transformations y equals f(x) plus a, f(x plus a), f(ax) and af(x).

What this dot point is asking

OCR wants you to sketch curves of standard functions (polynomials, the reciprocal $y = 1/x$ and $y = 1/x^2$, and translations of them), identify asymptotes and intercepts, find points of intersection of two graphs, and apply the four transformations $y = f(x) + a$, $y = f(x + a)$, $y = f(ax)$ and $y = af(x)$, including to unfamiliar functions given only by a sketch.

The answer

Sketching polynomials

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 reciprocal function

$y = \dfrac{1}{x}$ has two branches, a vertical asymptote at $x = 0$ and a horizontal asymptote at $y = 0$. The curve $y = \dfrac{1}{x^2}$ sits entirely above the $x$-axis with the same asymptotes. Translations move the asymptotes with the curve.

Points of intersection

To find where two curves meet, set their equations equal and solve. The number of solutions is the number of intersection points; the discriminant can tell you how many without solving.

The four transformations

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.

Examples in context

Transforming an unknown function

Combining a stretch and a reflection

When several transformations are applied, work from the inside out for changes to $x$ and treat the outside changes to $y$ separately. A negative scale factor combines a stretch with a reflection: $y = -2f(x)$ is a vertical stretch of factor $2$ followed by a reflection in the $x$-axis.

Reading information from a sketch

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.

Try this

Q1. Describe the transformation taking $y = f(x)$ to $y = f(x) - 4$. [1 mark]

• Cue. Translation by $\begin{pmatrix} 0 \\ -4 \end{pmatrix}$ (down 4).

Q2. The graph $y = \sin x$ is stretched to give $y = \sin 2x$. State the new period. [2 marks]

• Cue. Horizontal stretch factor $\tfrac{1}{2}$, so the period halves from $2\pi$ to $\pi$.