Secondary 5 Rational Functions Worksheet

All 8 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Finding the Rule of a Rational Function

   A rational function has the asymptotes $x=2$ and $y=-3$, and its curve passes through the point $(4,1)$.

   1. Find its rule in the form $f(x)={\displaystyle \frac{a}{x-h}}+k$.
   2. Give the coordinates of the two intercepts of the curve.
2. Q2Graphing a Rational Function

   Consider $f(x)={\displaystyle \frac{6}{x-1}}+2$.

   1. Give the equations of the two asymptotes and the coordinates of the two intercepts.
   2. Sketch both branches on the grid below, using at least two extra plotted points per branch.
   3. Give the domain and the range of $f$.

   A blank Cartesian grid for this question is on the printable PDF.

3. Q3Properties of the Rational Function

   Let $f(x)={\displaystyle \frac{4}{x-3}}+2$.

   1. Give the two asymptotes, the domain and the range.
   2. Find the zero and the $y$-intercept.
   3. Give the intervals where $f$ is positive and where it is negative.
4. Q4Solving Problems Involving Rational Functions

   A robotics club buys a 3D printer for $$1800$; each printed part then costs $$4$ in filament. After $n$ parts, the average cost per part is $$C(n)=\frac{1800+4n}{n},n>0.$$

   1. What is the average cost per part after $150$ parts?
   2. How many parts must the club print for the average cost to drop below $$10$?
   3. What is the horizontal asymptote of $C$, and what does it mean here?
5. Q5Switching from the Standard to the General Form of a Rational Function
   1. Write $f(x)={\displaystyle \frac{3}{x-4}}+5$ in the general form ${\displaystyle \frac{ax+b}{cx+d}}$.
   2. Write $g(x)={\displaystyle \frac{2x+11}{x+3}}$ in standard form and give its asymptotes.
6. Q6The Inverse of the Rational Function

   Let $f(x)={\displaystyle \frac{3}{x-4}}+2$. Find the rule of $f^{-1}$ and give the equations of the asymptotes of $f$ and of $f^{-1}$.

7. Q7The Rational Function

   Here “rational function” means a function whose rule is a quotient of polynomials in which the variable actually appears in the denominator. For each rule, state whether it defines a rational function, and whether its graph has a vertical asymptote. Justify.

   1. $a(x)={\displaystyle \frac{5}{2x-7}}$
   2. $b(x)={\displaystyle \frac{x^2-4}{x-2}}$
   3. $c(x)={\displaystyle \frac{x+1}{4}}$
8. Q8The Role of the Parameters in a Rational Function in Standard Form

   Let $f(x)={\displaystyle \frac{-4}{x+3}}+2$.

   1. Give the equations of the two asymptotes, the domain and the range.
   2. Is each branch increasing or decreasing? Justify with the sign of $a$.
   3. Find the zero of $f$.

The 7 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

Standard form is a set of instructions, not a formula to memorise

Almost every question on this sheet lives in one shape:

Once a rule is in that shape, four of the answers are already written down. The denominator cannot be zero, so x = h is the vertical asymptote and the domain is every real number except h. The fraction a ⁄ (x − h) is a non-zero constant divided by something, so it is never zero itself — which means f(x) can be anything except k. That is the horizontal asymptote, and it is why the range is every real number except k. Q2, Q3 and Q8 are all asking you to perform that reading.

Why you never set the numerator to zero

Everywhere else in the course, "find the zero" means "set the numerator to zero". Here that reflex produces nonsense. In Q3 the numerator is the constant 4, and 4 = 0 is simply false — the fraction 4 ⁄ (x − 3) is never zero for any x. The curve reaches the x-axis only because of the + k outside. So the equation you actually solve is the fraction equalling −k. With a rule like 6 ⁄ (x + 1) − 3:

That "never zero" fact is doing double duty. It is also the reason the horizontal asymptote is never crossed: f(x) = k would force a ⁄ (x − h) = 0, which cannot happen while a ≠ 0. A curve in this family approaches y = k from one side on each branch and stays there.

Sign intervals: three pieces, and the asymptote is in none of them

Q3(c) asks where f is positive and where it is negative. The zero and the vertical asymptote cut the number line into three intervals; inside each one the sign cannot change, so a single test value settles it. Take f(x) = 2 ⁄ (x − 5) + 1, whose zero is at x = 3 and whose asymptote is x = 5: test x = 0, x = 4 and x = 6, and the pattern is fixed.

Increasing or decreasing: the sign of a does the opposite of what you expect

Q8(b) asks you to justify the variation from the sign of a, and the honest answer surprises most students. Follow one branch as x increases. Then x − h increases, so 1 ⁄ (x − h) gets smaller. Multiply by a positive a and it is still shrinking, so both branches decrease. Multiply by a negative a and the order reverses, so both branches increase.

Sketching both branches (Q2)

Draw the two asymptotes as dashed lines first. They are the scaffolding, and a sketch built on them is almost impossible to get badly wrong. Then place the two intercepts, which you already computed in part (a), and add the extra points the question asks for.

Choosing those extra points is the small skill worth having: pick x values that make x − h a divisor of a, and the coordinates come out as whole numbers. For Q2's rule, values of x sitting 1, 2, 3 or 6 units away from the asymptote do exactly that — on both sides, so you get points for both branches out of the same idea. With a > 0 the right-hand branch sits above y = k and the left-hand branch below it; with a < 0 they swap. Sketch a curve that hugs both dashed lines without ever touching them.

Standard form ↔ general form (Q5)

Q5 runs the conversion in both directions, and each direction is one idea.

Standard to general is just addition of fractions: put the + k over the same denominator and combine. 3 ⁄ (x − 4) + 5 becomes [3 + 5(x − 4)] ⁄ (x − 4), then expand the top.

General to standard is division: rewrite the numerator as a multiple of the denominator, plus whatever is left over. For (4x + 9) ⁄ (x + 2), ask how many times x + 2 fits into 4x + 9 — four times, using up 4x + 8 and leaving 1:

Now the asymptotes are readable: x = −2 and y = 4. If the leading coefficient of the denominator is not 1, the multiple is a fraction — that is allowed, and it is where the arithmetic gets careless.

The inverse, and the structural check that makes it free (Q6)

The mechanics are the usual ones: swap x and y, then isolate y. Starting from y = a ⁄ (x − h) + k you write x = a ⁄ (y − h) + k, move k across, and flip both sides of x − k = a ⁄ (y − h) to reach y − h = a ⁄ (x − k).

Finish with one numerical check: pick any convenient x, compute f(x), then feed that output into your f⁻¹ and confirm you land back where you started.

Asymptote or hole? (Q7)

Q7 gives you three rules and asks two separate questions about each: is it a rational function, and does its graph have a vertical asymptote. They are separate, and that is the whole point.

A rule counts as rational here when the variable genuinely appears in a denominator. A rule like (x + 1) ⁄ 4 only looks like a fraction; dividing by the constant 4 is multiplying by 1 ⁄ 4, so it is a first-degree polynomial function, defined everywhere, with no excluded value and no asymptote.

Average-cost models (Q4)

Q4 is the applied shape this topic almost always takes on an exam: a fixed cost paid once, a variable cost paid per unit, and a question about the average. Divide the total by the number of units and then split the fraction into two pieces:

That is standard form with a = F, h = 0 and k = r. Reading it that way answers part (c) before you calculate anything: the horizontal asymptote is y = r, the per-unit cost on its own, and the average cost falls toward it without ever reaching it, because the fixed cost is always being spread over some finite number of units.

Two habits for part (b) and its relatives. First, when you multiply both sides of the inequality by n, say out loud why the sense does not flip — it is because the context forces n > 0, not because inequalities are usually well behaved. Second, respect the context's domain: a number of printed parts is a positive whole number, so an algebraic answer of n > 300 is reported as "from the 301st part onward", not as an interval of reals.

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Getting the most out of it

Draw the dashed lines before you draw anything else

On every graphing question, put the two asymptotes on the grid first, as dashed lines, labelled with their equations. A branch drawn into that frame lands in the right region automatically; a branch drawn freehand and then "fitted" to asymptotes almost never does.

Convert to standard form before you answer any question about a rule

If a rule arrives as (ax + b) ⁄ (cx + d), do the division first, even when the question has not asked for it. Every property you might be asked for — asymptotes, domain, range, variation, the inverse — is visible in standard form and hidden in the general one.

Check every rule you build with the point you were given

Q1, Q5 and Q6 all end with a rule you constructed rather than one you were handed. Substituting the original point back in turns a guess into a verified answer, and it takes one line. Build the habit here and it pays for itself on the exam, where nobody tells you a sign is wrong.

Common questions