MHF4U · Unit 2 test review

Rational Functions: Test Review

Everything for the Unit 2 test on one page: the key facts from all 4 lessons, the mistakes that cost the most marks, and a practice test that points you back to the lessons you need.

Key facts

Reciprocal Functions

The reciprocal of $f$ is $\frac{1}{f ( x )}$ , defined for every $x$ with $f (x) \neq = 0$ — always state that restriction.
Each zero of $f$ becomes a vertical asymptote of $\frac{1}{f ( x )}$ .
As $∣ f (x) ∣ \to \infty$ , $\frac{1}{f ( x )} \to 0$ : the horizontal asymptote is $y = 0$ , and the graph never touches it.
$\frac{1}{f ( x )}$ has exactly the same sign as $f (x)$ on every interval where it is defined.
Invariant points occur where $f (x) = 1$ or $f (x) = - 1$ , never where $f (x) = 0$ .
A minimum of $f$ with a non-zero value lines up with a local maximum of $\frac{1}{f ( x )}$ at the same $x$ -value.

Restrictions, Asymptotes and Holes

Restrictions are the zeros of the original denominator, found before any cancelling, and they never go away.
A denominator factor that does not cancel gives a vertical asymptote; a factor that cancels gives a hole.
The $y$ -value of a hole comes from substituting into the simplified function, never the original.
Degrees: numerator lower gives $y = 0$ ; equal gives $y = \frac{a}{b}$ from the leading coefficients; numerator one higher gives an oblique asymptote.
The oblique asymptote is the quotient of the division — discard the remainder, since it tends to $0$ .
A graph may cross a horizontal or oblique asymptote in the middle; only its end behaviour is guaranteed.

Sketching Rational Functions

$x$ -intercepts are the zeros of the simplified numerator; the $y$ -intercept is $f (0)$ when $0$ is in the domain.
Critical values for the sign chart are the $x$ -intercepts and the vertical asymptotes.
The sign of $f$ on the interval beside a vertical asymptote tells you whether that branch goes to $+ \infty$ or $- \infty$ .
Near a hole the curve is smooth and finite; only one point is missing.
To test whether the curve meets its horizontal asymptote $y = k$ , solve $f (x) = k$ .

Rational Equations and Inequalities

State the restrictions — every zero of every denominator — before solving anything.
To solve an equation, multiply every term by the LCD, then solve the resulting polynomial equation.
A root that equals a restriction is extraneous: reject it and say so. All roots rejected means "no solution".
Never multiply or cross-multiply an inequality by an expression containing the variable — its sign is unknown.
For an inequality, rewrite it as a single rational expression compared to $0$ , then build a sign chart.
Critical values come from the numerator (square brackets allowed) and the denominator (always round brackets).

Where marks get lost

Putting the vertical asymptote at the vertex of the parabola instead of at its zeros. The vertex controls the turning point of the reciprocal; only the zeros create asymptotes. (Reciprocal Functions)
Marking invariant points where $f (x) = 0$ . A zero is precisely where the reciprocal is undefined — invariant points need $f (x) = \pm 1$ . (Reciprocal Functions)
Assuming the reciprocal of a quadratic always has two vertical asymptotes. Check the discriminant: it can have two, one, or none. (Reciprocal Functions)
Flipping the sign as well as the value. The reciprocal of a negative number is still negative, so a branch where $f$ is below the axis stays below the axis. (Reciprocal Functions)
Writing the domain as "all real numbers" and forgetting to exclude the zeros of the original function — the restriction is worth marks on its own. (Reciprocal Functions)
Cancelling first and then reading the restrictions off the simplified expression. The restriction $x \neq = 3$ in $\frac{( x - 3 ) ( x + 2 )}{( x - 3 ) ( x + 3 )}$ disappears from view but not from the function. (Restrictions, Asymptotes and Holes)
Calling a cancelled factor a vertical asymptote. If it cancels it is a hole, and the graph is finite there. (Restrictions, Asymptotes and Holes)
Substituting the hole's $x$ -value into the original function, getting $\frac{0}{0}$ , and concluding the point does not exist. Use the simplified function to get its height. (Restrictions, Asymptotes and Holes)
Using constant terms instead of leading coefficients for the horizontal asymptote — $\frac{3 x ^{2} - 1}{2 x ^{2} + 5}$ has asymptote $y = \frac{3}{2}$ , not $y = - \frac{1}{5}$ . (Restrictions, Asymptotes and Holes)
Believing a curve can never cross its horizontal asymptote. It often does near the middle; the asymptote only describes what happens far out. (Restrictions, Asymptotes and Holes)
Letting the curve touch or cross a vertical asymptote. The function has no output there, so the graph must stay strictly on one side of the line. (Sketching Rational Functions)
Testing only one side of a vertical asymptote. The two sides can go the same direction or opposite directions — the sign chart decides, not intuition. (Sketching Rational Functions)
Drawing a hole as a break or a vertical asymptote. A hole is a single missing point on an otherwise smooth curve. (Sketching Rational Functions)
Marking an $x$ -intercept where the numerator is zero but the value is a restriction. That point is a hole, so the graph does not cross there. (Sketching Rational Functions)
Bending the curve away from a horizontal asymptote just because it already crossed it. Crossing in the middle is allowed; the asymptote only governs the far ends. (Sketching Rational Functions)
Solving correctly and then reporting a root that was restricted. Always compare every root to the restriction list before writing the final answer. (Rational Equations and Inequalities)
Multiplying an inequality by $(x - 3)$ as though it were positive. When $x < 3$ that expression is negative and the inequality sign must flip, so the method silently loses half the solution set. (Rational Equations and Inequalities)
Cross-multiplying $\frac{a}{b} < \frac{c}{d}$ . This is the same illegal move as multiplying by a variable expression. (Rational Equations and Inequalities)
Leaving the denominator's zeros off the sign chart. The expression can change sign there (it does at any odd-multiplicity zero) even though it is never equal to zero there. (Rational Equations and Inequalities)
Using a square bracket at a restricted value, for example writing $[2, \infty)$ when $x = 2$ makes the denominator zero. Restricted endpoints are always round. (Rational Equations and Inequalities)

Practice test

24 questions from across the unit. Each one shows the lesson it comes from.

Reciprocal Functions
1
What is the vertical asymptote of $y = \frac{1}{3 x + 9}$ ?
Reciprocal Functions
2
How many vertical asymptotes does $y = \frac{1}{x ^{2} + 9}$ have?
Reciprocal Functions
3
State the horizontal asymptote of the reciprocal of any linear function.
Reciprocal Functions
4
On the interval $x < - 2$ , where does the graph of $y = \frac{1}{x + 2}$ lie?
Reciprocal Functions
5
For $y = \frac{1}{x ^{2} - 4}$ , one invariant point has $y = 1$ and a positive $x$ -coordinate. What is that $x$ -coordinate? Round to three decimal places.
Reciprocal Functions
6
Match each feature of f with what it produces on the graph of 1/f.
A zero of f
f(x) = 1 or f(x) = −1
A minimum of f
Very large values of f
Restrictions, Asymptotes and Holes
7
What is the horizontal asymptote of $f (x) = \frac{4 x ^{2} - x}{2 x ^{2} + 5}$ ?
Restrictions, Asymptotes and Holes
8
Which describes the end behaviour of $f (x) = \frac{x ^{3} + 1}{x - 1}$ ?
Restrictions, Asymptotes and Holes
9
Find the slope of the oblique asymptote of $f (x) = \frac{2 x ^{2} + 3 x - 1}{x + 1}$ .
Restrictions, Asymptotes and Holes
10
After simplifying $\frac{x ^{2} - 25}{x ^{2} - 3 x - 10}$ to $\frac{x + 5}{x + 2}$ , what is true at $x = 5$ ?
Restrictions, Asymptotes and Holes
11
Order the steps for classifying the restrictions of a rational function:
1. 1Substitute each hole's x-value into the simplified function to find its height
2. 2Label the cancelling factors as holes and the rest as vertical asymptotes
3. 3Check each denominator factor for a matching numerator factor
4. 4Factor the numerator and denominator completely
5. 5Set the original denominator to zero to list every restriction
Restrictions, Asymptotes and Holes
12
How many vertical asymptotes does $f (x) = \frac{x + 2}{( x + 2 ) ( x - 1 ) ( x + 5 )}$ have?
Sketching Rational Functions
13
What is the $y$ -intercept of $f (x) = \frac{2 x - 6}{x + 3}$ ?
Sketching Rational Functions
14
What are the $x$ -intercepts of $f (x) = \frac{x ^{2} - x - 6}{x - 3}$ ?
Sketching Rational Functions
15
Near the vertical asymptote of $f (x) = \frac{3}{( x - 2 ) ^{2}}$ , how does the graph behave?
Sketching Rational Functions
16
For $f (x) = \frac{x - 1}{x + 2}$ , is $f$ positive or negative on the interval $(- 2, 1)$ ?
Sketching Rational Functions
17
Order the sketching procedure for a rational function:
1. 1Build a sign chart from the intercepts and vertical asymptotes
2. 2Find the vertical asymptote(s) and the horizontal or oblique asymptote
3. 3Draw the branches through the known points and towards the asymptotes
4. 4Factor the numerator and denominator
5. 5State the restrictions and simplify
6. 6Find the x- and y-intercepts
Sketching Rational Functions
18
Does the graph of $f (x) = \frac{2 x}{x ^{2} + 1}$ ever cross its horizontal asymptote $y = 0$ ?
Rational Equations and Inequalities
19
Solve $\frac{5}{x - 3} = 1$ .
Rational Equations and Inequalities
20
Solve $\frac{1}{x} + \frac{1}{2} = \frac{3}{4}$ .
Rational Equations and Inequalities
21
What is the solution set of $\frac{x - 2}{x + 1} < 0$ ?
Rational Equations and Inequalities
22
Why is it wrong to multiply both sides of $\frac{5}{x - 3} > 2$ by $x - 3$ ?
Rational Equations and Inequalities
23
In the solution of $\frac{x - 4}{x + 3} \geq 0$ , which brackets are correct at $x = - 3$ and $x = 4$ ?
Rational Equations and Inequalities
24
Match each situation to the correct action.
A root equals a restricted value
Every root is rejected
Solving an inequality
Endpoint is a zero of the denominator
Endpoint is a zero of the numerator in a ≥ inequality

Endless drill

Fresh numbers every time, for the calculation questions in this unit.

Endless drillnew numbers every time

Generating a problem…