Sketching is a reasoning tool: the number of solutions of \(f(x) = g(x)\) is the number of intersections of the graphs, and the sign of a function is read from a sketch. Know the standard shapes and how roots, repeated roots and the leading coefficient control a polynomial's picture.
Contents: Standard graphs (MM8.1, M4.15)Roots and intersections (MM8.6, MM8.7)Must-knowTrapsReal questions
Standard graphs (MM8.1, M4.15)
- Lines, parabolas, cubics (an S-shape; \(+x^3\) rises to the right), \(y = \frac1x\) (two branches, asymptotes both axes), \(y = a^x\) (through \((0, 1)\)), \(y = \log_a x\) (through \((1, 0)\), only \(x > 0\)), \(y = \sqrt x\) (only \(x \geq 0\)), \(y = |x|\) (V-shape), \(\sin\), \(\cos\), \(\tan\).
- \(y = |f(x)|\): reflect the parts below the axis upwards. \(|x^2 - 4|\) is a W with minima \(0\) at \(\pm2\) and a local maximum \(4\) at \(0\).
- Reflection in \(y = x\) gives the inverse: \(\log_2 x \leftrightarrow 2^x\).
Roots and intersections (MM8.6, MM8.7)
- A polynomial of degree \(n\) has at most \(n\) real roots; an odd degree guarantees at least one.
- A factor \((x - a)^2\) makes the graph touch the axis at \(a\); \((x - a)^3\) crosses with a flat point.
- \(y\)-intercept: put \(x = 0\). The sign of the leading coefficient fixes the behaviour as \(x \to \pm\infty\).
- Solutions of \(f(x) = k\) = intersections with the horizontal line \(y = k\): slide the line to count how the number changes (e.g. exactly three solutions of \(|x^2 - 4| = k\) only when \(k = 4\)).
- Line meets curve: substitute and use the discriminant (or a sketch when no algebra is possible, e.g. \(\sin x = \frac{x}{10}\)).
Must-know (no formula booklet)
- degree \(n\): at most \(n\) roots; odd degree: at least \(1\)
- \((x-a)^2\) factor: touches
- intersections of \(y = f(x)\) and \(y = g(x)\) = solutions of \(f(x) = g(x)\)
- \(y = |f(x)|\) reflects negative parts up
Classic traps
- \(x^2 + cx - 1 = 0\) always has two roots (discriminant \(c^2 + 4 > 0\)): \(y = x + c\) always meets \(y = \frac1x\) twice.
- \(\sqrt{4 - x} = \sqrt{-(x - 4)}\): reflect first, then translate \(4\) right (the other order gives \(\sqrt{-x - 4}\)).
- A cubic with three real roots must have two turning points, but two turning points do not guarantee three roots.
Worked example
Q. How many real solutions has \(x^3 = 3x + 1\)?
A. \(y = x^3 - 3x - 1\) has turning values \(y(-1) = 1 > 0\) and \(y(1) = -3 < 0\), so three crossings.
How it has been examined
2022 · Paper 1Question 20intersections of y = |x^2 - a^2| and y = a^2 |x - 1|question papersit online