SL 2.4 Key features of graphs

Reading a graph like a map

Every graph question asks some version of "where is the interesting stuff?". The interesting stuff is a short list:

• Intercepts: where the curve crosses the axes. Roots (zeros) are the \(x\)-intercepts, solutions of \(f(x) = 0\).
• Maximum and minimum points: the highest or lowest points, either locally (a hill or valley) or over the whole domain.
• Symmetry: for a parabola, the vertical axis through the vertex; for other curves, symmetry in the \(y\)-axis or the origin.
• Asymptotes: lines that the curve approaches but does not reach as it heads off to infinity.

Asymptotes of simple rational functions

For \(f(x) = \dfrac{ax + b}{cx + d}\):

• Vertical asymptote where the denominator is zero, \(x = -\frac{d}{c}\), because the output blows up there.
• Horizontal asymptote \(y = \frac{a}{c}\), because for huge \(x\) the constants \(b\) and \(d\) hardly matter and \(f(x) \approx \frac{ax}{cx}\).

Worked example

Find the key features of \(f(x) = \dfrac{2x + 1}{x - 3}\).

• Vertical asymptote: \(x - 3 = 0\), so \(x = 3\).
• Horizontal asymptote: \(y = \frac{2}{1} = 2\).
• \(x\)-intercept: numerator zero, \(2x + 1 = 0\), so \(\left(-\frac12, 0\right)\).
• \(y\)-intercept: \(f(0) = \frac{1}{-3}\), so \(\left(0, -\frac13\right)\).

So the graph has two branches, one to the right of \(x = 3\) above \(y = 2\) and one to the left below it. The left branch passes through both intercepts.

Intersections with technology

To solve \(f(x) = g(x)\) when algebra is hard, graph both and use the intersect tool. The \(x\)-coordinates of the intersection points are the solutions. For example, \(e^x = x + 2\) has two solutions, near \(x \approx -1.84\) and \(x \approx 1.15\), which no algebraic method gives exactly. You can also graph \(y = f(x) - g(x)\) and find its zeros: same answers.

Maxima and minima in context

A local maximum is higher than nearby points; the global maximum is the highest on the whole domain and may sit at an endpoint. For a model like profit \(P(x)\), the maximum value and where it happens are often both requested: give the coordinate and its meaning.

Common mistakes

• Saying a curve can never cross a horizontal asymptote. It can, at finite \(x\); the asymptote describes behaviour far away.
• Giving only the \(x\)-coordinate when a point is asked for.
• Missing an endpoint maximum on a restricted domain.
• Writing an asymptote as a number ("3") instead of an equation (\(x = 3\)).

Exam tips

• State asymptotes as equations and draw them dashed.
• On Paper 2, give GDC results to 3 significant figures unless exact values are possible.
• When asked "how many solutions", sketch both sides and count crossings; make sure the window shows them all.