Rational functions

Asymptotes, intercepts, sketch, range

In 12 of 20 recent exams
1% of the marks
Paper 1 or 2
1 to 5 marks

How it is set in the exam

Paper 1 (no calculator), 1–5 marks per part: a short question of 5–7 marks (the asymptotes, the intercepts, a sketch), or the first parts of a Section B question that goes on to the inverse (or where $f$ meets $f^{-1}$ ), a transformation, the derivative, a line that meets the graph once, or a second graph. Very common (12 of 20 exam versions; 27 of its 34 question parts are on Paper 1 (no calculator)). On Paper 2 (calculator) (3 versions) the parts are the same and still exact. The counts at the top of this guide (35 parts, 13 versions) add the one part of Find a rational function's constants.

Spot it in the exam

Exam phrases to spot (5)

“write down the equation of the vertical asymptote”
“write down the equations of the asymptotes of the graph of $f$ ”
“find the coordinates of the axis intercepts”
“sketch the graph of $f$ , showing its asymptotes and axis intercepts”
“write down the range of $f$ ”

In disguise (6)

the coefficients are letters: "find, in terms of $k$ , the equations of the asymptotes" (Example 1.3)
a factorised or reordered formula, such as $\frac{3(2 - x)}{x + 1}$ or $\frac{1 + 4x}{3 - x}$ : the horizontal asymptote is the ratio of the $x$ -coefficients, whatever order the terms are in (Example 1.4)
a negative $x$ -coefficient in the denominator, such as $\frac{5x + 2}{1 - 2x}$ (Example 1.2)
the formula is already in the form $p + \frac{q}{x - h}$ : read both asymptotes straight off (The form p + q/(x − h))
one more $x$ -value is left out than the asymptote's: an open circle, and one more value missing from the range (Example 1.3)
a sketch of $y = \frac{1}{g(x)}$ for a given $g(x) = \frac{ax + b}{cx + d}$ : flip the fraction to $\frac{cx + d}{ax + b}$ , another function of this shape, and use the same Steps (no AHL method is needed). The $x$ -value left out of the domain of $g$ stays out, so where the flipped fraction would meet the $x$ -axis there is an open circle, and $0$ leaves the range (one markscheme also accepted an intercept drawn there)

Variants

asked for = one or both asymptotes, one or both intercepts, a sketch (worth 1 to 5 marks), the range; coefficients as numbers or letters; an extra excluded point; followed by a count of the solutions of $f(x) = g(x)$ for a given quadratic $g$ , from the same sketch: that part is Solve graphically: intersections, how many solutions, above or below a level, worked here in Example 1.4.

Looks like it, but isn't (8)

the formula for $f^{-1}$ or its domain (the domain is the range of $f$ , found here; Example 1.2(d) chains into it), or showing that $f^{-1} = f$ : Find the inverse function, with its domain
where the graphs of $f$ and $f^{-1}$ meet (for a rational $f$ , a quadratic): The graphs of f and f⁻¹: reflect in y = x, find where they meet
the transformations that take the reciprocal graph to this one: Name the transformation: describe it or find its constants
the graph is transformed first, and the transformed graph's asymptotes or range are asked for: Transform a graph: new equation, image points, sketch, range
the constants are unknown and features are given: Find a rational function's constants from its features
a sloping line $y = mx + c$ ( $m \ne 0$ ) and the graph: where they meet, or tangency: Two graphs: tangent, meet, don't meet (clear the denominator: a quadratic)
an inequality such as $f(x) > 0$ or $f(x) \le 2$ , or the number of solutions of $f(x) = g(x)$ , read off the sketch: Solve graphically: intersections, how many solutions, above or below a level (Example 1.4 works a count after the sketch)
the asymptote of an exponential graph: Exponential and log graphs: intercepts, asymptotes, domain, sign, range; the value a model in a context settles at: Rate of change in context: find it and interpret it

Worked example

The standard chain: asymptotes, intercepts, sketch

Paper 1 (no calculator)

Question

Let $f(x) = \dfrac{3x - 5}{x + 4}$ , for $x \ne -4$ .
(a) Write down the equations of the asymptotes of the graph of $f$ . [2]
(b) Write down the coordinates of the points where the graph of $f$ meets the axes. [2]
(c) Sketch the graph of $f$ , showing its asymptotes and the points found in (b). [3]

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See it move: Move the asymptotes, the formula follows

$f(x) = \dfrac{3x - 5}{x + 4}$ . The vertical asymptote is where the bottom is 0: $x + 4 = 0$ , so $x = -4$ .
The horizontal asymptote is the ratio of the $x$ -coefficients, $\frac{3}{1}$ : $y = 3$ . The constants $-5$ and $4$ play no part in it.
The intercepts: $f(0) = \frac{-5}{4}$ , so $\left(0, -\frac{5}{4}\right)$ ; the top is 0 at $3x - 5 = 0$ , so $\left(\frac{5}{3}, 0\right)$ .
The intercepts are bottom-right of where the asymptotes cross, so one branch is there and the other opposite: the branches are top-left and bottom-right. Written as $3 - \frac{17}{x + 4}$ , the sign of $-17$ says the same.
A test value: $f(-7) = \frac{26}{3}$ , above $y = 3$ , on the left branch. The curve never takes the value $3$ : the range is $y \ne 3$ .
(a) $x = -4$ and $y = 3$ , each written as an equation. (b) $\left(0, -\frac{5}{4}\right)$ and $\left(\frac{5}{3}, 0\right)$ , as coordinates. (c) Both intercepts are bottom-right, so the branches are bottom-right and top-left; $f(-5) = 20 > 3$ checks it.

Which method?

Which pair of quarters holds the branches?

an intercept (almost always)
Use
the quarter it is in holds one branch; the opposite quarter holds the other
Example
$\frac{3x - 5}{x + 4}$ : $\left(0, -\frac{5}{4}\right)$ is right of $x = -4$ and below $y = 3$ , so bottom-right, and the other branch is top-left
no intercept (as for $\frac{k}{x}$ ), or as a check
Use
a test value, on either side of the vertical asymptote
Example
$-\frac{2}{x}$ : $x = 1$ gives $-2$ , below $y = 0$ , so bottom-right, and the other branch is top-left. As a check on any sketch: for $\frac{5}{x - 2}$ , $x = 3$ gives $5$ , top-right, which agrees with the $y$ -intercept $\left(0, -\frac{5}{2}\right)$ being bottom-left
the form $p + \frac{q}{x - h}$
Use
the sign of $q$ : $q > 0$ top-right and bottom-left, as for $\frac{1}{x}$ ; $q < 0$ top-left and bottom-right
Example
$3 - \frac{4}{x - 1}$ : $q = -4 < 0$

You have	Use	Example
an intercept (almost always)	the quarter it is in holds one branch; the opposite quarter holds the other	$\frac{3x - 5}{x + 4}$ : $\left(0, -\frac{5}{4}\right)$ is right of $x = -4$ and below $y = 3$ , so bottom-right, and the other branch is top-left
no intercept (as for $\frac{k}{x}$ ), or as a check	a test value, on either side of the vertical asymptote	$-\frac{2}{x}$ : $x = 1$ gives $-2$ , below $y = 0$ , so bottom-right, and the other branch is top-left. As a check on any sketch: for $\frac{5}{x - 2}$ , $x = 3$ gives $5$ , top-right, which agrees with the $y$ -intercept $\left(0, -\frac{5}{2}\right)$ being bottom-left
the form $p + \frac{q}{x - h}$	the sign of $q$ : $q > 0$ top-right and bottom-left, as for $\frac{1}{x}$ ; $q < 0$ top-left and bottom-right	$3 - \frac{4}{x - 1}$ : $q = -4 < 0$

Steps

Vertical asymptote:
set the denominator equal to $0$ : $x = -\frac{d}{c}$ . Write it as an equation.
Horizontal asymptote:
$y = \frac{a}{c}$ , the ratio of the $x$ -coefficients with their signs. If the formula is factorised or reordered, find $a$ and $c$ first (expand the top or the bottom if you need to).
Intercepts:
$f(0) = \frac{b}{d}$ ; numerator $= 0$ for the $x$ -intercept. Give them as coordinates.
Sketch:
dashed asymptotes labelled with their equations; the intercepts labelled; one branch in each of two opposite quarters, chosen by the table above; both branches approaching both asymptotes. An open circle only at an $x$ -value that the stated domain leaves out.
Domain and range:
$x \ne -\frac{d}{c}$ and $y \ne \frac{a}{c}$ . For each extra excluded $x_0$ , remove $f(x_0)$ from the range as well.
Check:
every intercept lies on a branch you drew, and a test value lands in the quarter you chose.
A follow-on count, $f(x) = g(x)$
(Solve graphically: intersections, how many solutions, above or below a level): sketch both graphs. On each side of the vertical asymptote, first compare the ranges: each branch of $f$ lies entirely above or entirely below $y = \frac{a}{c}$ , and the parabola never goes past the $y$ -value of its vertex, so on one side they may not be able to meet at all (Example 1.4). Otherwise compare them at a few $x$ -values: the intercepts of both graphs, the other graph's vertex, next to the asymptote, where $f$ is very large, positive or negative, and far out, where $f$ is close to its horizontal asymptote and the parabola is very large, positive or negative. Each change in which graph is on top means at least one meeting. Between two test points, one graph rising while the other falls means at most one meeting there. If the shapes don't settle it (both falling, say), use the degree: clearing the denominator gives a cubic, which has at most three roots, so three changes of order means exactly three meetings. Don't solve the cubic.

Common mistakes

An asymptote as a number, or with "≠"
write $x = -4$ , not " $-4$ ", "VA $= -4$ " or " $x \ne -4$ " (that is the domain). Most markschemes simply require each answer to be an equation. A few have taken only one mark when the same wrong notation was used on both asymptotes, and both marks when two different wrong forms were mixed. Don't count on that leniency.
The horizontal asymptote from the wrong numbers
use the $x$ -coefficients. In $\frac{3(2 - x)}{x + 1}$ the top's is $-3$ , so $y = -3$ ; in $\frac{1 + 4x}{3 - x}$ it is $y = \frac{4}{-1} = -4$ , not $\frac{1}{3}$ .
Dropping the sign of $c$
$\frac{5x + 2}{1 - 2x}$ has $y = -\frac{5}{2}$ , not $\frac{5}{2}$ .

More mistakes (8)

The intercepts from the wrong part
the $x$ -intercept comes from the numerator; the denominator gives the asymptote. The $y$ -intercept is $\frac{b}{d}$ , not $\frac{a}{c}$ .
A branch that crosses the horizontal asymptote or turns back
neither happens for $\frac{ax + b}{cx + d}$ . Each branch approaches the asymptote and has no turning point.
Both branches on one side of the vertical asymptote,
or in neighbouring quarters: they are always in opposite quarters.
Unlabelled sketches
some markschemes accept asymptotes and intercepts drawn in roughly the right places; others give a separate mark for each asymptote labelled with its equation and each intercept labelled with its coordinates. Label every one: that scores under every scheme.
The range written with $x$
" $x \ne -\frac{5}{2}$ " describes inputs, so it is a domain. The range of $g$ in Example 1.2 is $y \ne -\frac{5}{2}$ (or $g(x) \ne -\frac{5}{2}$ ); the same number with $x$ is the domain of $g^{-1}$ .
A range joined by "and"
" $y < 3$ and $y > 3$ " describes no value. Write $y \ne 3$ , or $y < 3$ or $y > 3$ .
An extra gap in the domain
draw an open circle only where the question's domain leaves out a value (never a hole made by cancelling), and remove that point's $y$ -value from the range too (Example 1.3).
Counting one meeting per branch
in "how many solutions": compare the two graphs on each side of the asymptote (Example 1.4). A count read off a wrong sketch keeps its method mark but loses the answer mark, even when the number is right.

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