Find a rational function's constants

From its features

In 1 of 20 recent exams
Paper 1, no calculator

How it is set in the exam

Paper 1 (no calculator), 3 marks in the one instance ((M1) for substituting a point, (M1) for solving the pair, A1 for both values), as the first part of a question whose later parts use the function (there, the transformations from $\frac{1}{x}$ ). Rare (1 of 20 exam versions: two points on the graph, two unknowns). The version with a given asymptote had no 2021–26 instance; it uses the same facts.

Spot it in the exam

Exam phrases to spot (3)

“the graph passes through two given points; find the values of the two constants”
“find the constants, given that the graph of $f$ has asymptotes $x = 5$ and $y = -2$ ”
“the graph passes through … and its horizontal asymptote is $y = -3$ ”

In disguise (2)

the asymptotes are given as a domain and a range: "the domain of $f$ is given as $x \ne 6$ , the range as $y \ne -2$ "
a point is given as an intercept: " $f(0) = 3$ ", or "the $x$ -intercept is …"

Looks like it, but isn't (3)

the letters stay as letters, and the asymptotes are asked for in terms of them: Rational functions: asymptotes, intercepts, sketch, range (Example 1.3)
a constant from the fact that $f$ is its own inverse: Find the inverse function, with its domain
the transformations from $\frac{1}{x}$ are asked for, or the constants in them: Name the transformation: describe it or find its constants; if the transformations are given and the new formula is asked for: Transform a graph: new equation, image points, sketch, range

Worked example

A horizontal asymptote and a point

Paper 1 (no calculator)

Question

Let $f(x) = \dfrac{ax - 5}{bx + 3}$ , for $x \ne -\frac{3}{b}$ , where $a$ and $b$ are constants. The graph of $f$ has a horizontal asymptote $y = -3$ and passes through the point $(-1, 1)$ .
(a) Find the value of $a$ and the value of $b$ . [5]
(b) Write down the equation of the vertical asymptote of the graph of $f$ . [1]

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Steps

Each fact gives one equation. The table's letters are positions in $\frac{ax + b}{cx + d}$ : match them to the question's letters first (in Example 2.1 the question's $b$ is in the table's $c$ position).

Fact	Equation
vertical asymptote $x = h$	$-\frac{d}{c} = h$
horizontal asymptote $y = p$	$\frac{a}{c} = p$ , so $a = pc$ : a relation between two unknowns when both are letters
a point $(x_0, y_0)$ on the graph	$\frac{ax_0 + b}{cx_0 + d} = y_0$ ; multiply by the denominator: a linear equation
$y$ -intercept $y_0$	$\frac{b}{d} = y_0$
$x$ -intercept $x_0$	$ax_0 + b = 0$

Count: list the unknowns and the facts. Each unknown needs one fact: two unknowns, two facts; three unknowns, three facts. A fact that involves only one unknown (often a vertical asymptote, or $f(0)$ ) fixes it at once.
Write one equation per fact with the table. Clear every denominator.
Solve the linear equations by substitution or elimination. On , write the equations down, then use the GDC: solve simultaneous linear equations.
Check the function you found against each fact, and check that it is a genuine rational function (the top is not a multiple of the bottom: $ad \ne bc$ ).

Common mistakes

Taking the asymptote as $a$ itself
with the denominator $bx + 3$ , the horizontal asymptote is $\frac{a}{b}$ .
The vertical asymptote as $-d$
for $bx + 3$ it is $x = -\frac{3}{b}$ .
The gaps swapped
the value missing from the domain gives the vertical asymptote; the value missing from the range gives the horizontal one (Your turn 3).

More mistakes (2)

Multiplying only part of the denominator
$\frac{3a + b}{6 + c} = 2$ gives $3a + b = 2(6 + c) = 12 + 2c$ , not $12 + c$ . Bracket the whole denominator before multiplying.
Stopping at the constants when the function is asked for
write $f(x)$ out in full, and check it against every fact.

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