Rational functions · Grade 10 · 5 units
In Rational functions (Grade 10 · 5 units) you practise 7 skills: The domain and a missing point · Asymptotes · The derivative of one over a function · The quotient rule · Investigating a rational function · A tangent to a rational graph · Reading the graphs. A rational function is a fraction with a polynomial on the top and on the bottom. It is not defined where the denominator is 0. When the numerator is 0 there too, cancel the common factor and look at the reduced fraction: if its denominator is not 0 there any more, the graph has a point missing, and if it is still 0, there is a vertical asymptote. When only the denominator is 0, there is a vertical asymptote there: next to it the function grows or falls without bound.
Far from the origin the highest powers decide. When the numerator and the denominator have the same degree, the function approaches the ratio of the coefficients of the highest powers, and that is the horizontal asymptote. When the numerator has the greater degree, the values grow without bound in size and there is no horizontal asymptote. The derivative of one over x is minus one over x squared. For a general fraction use the quotient rule: the derivative of the top times the bottom, minus the top times the derivative of the bottom, all over the bottom squared. The derivative finds tangents, where the function increases and decreases, and its turning points. A point outside the domain splits an interval in two.
Tip: Before you differentiate, write down the domain: the points where the denominator is 0 are never part of the answer.
What you learn
The domain and a missing point
For example: Which value of x is not in the domain of the function f(x) = −7 / (x − 6) ?
Answer and solution
The answer: 6
The denominator x − 6 is 0 when x = 6 .
Asymptotes
For example: Find the vertical asymptote of the graph of f(x) = −5 / (x − 6) − 5 .
Answer and solution
The answer: 6
The denominator x − 6 is 0 at x = 6 , and the numerator is not 0. That is the vertical asymptote.
The derivative of one over a function
For example: Here is the function f(x) = −5 / x . Find f′(4) .
Answer and solution
The answer: 5/16
By the derivative of one over x, f′(x) = 5 / x² . Put it in: f′(4) = 5 / 4² = 5/16 .
The quotient rule
For example: Here is the function f(x) = (3x + 3) / (x − 2) . Find f′(0) .
Answer and solution
The answer: −9/4
By the quotient rule: f′(x) = (3(x − 2) − (3x + 3)) / (x − 2)² = −9 / (x − 2)² . Put in x = 0 : f′(0) = −9 / (−2)² = −9/4 .
Investigating a rational function
For example: What is the sign of the function f(x) = (x + 5) / (x − 6) for x < −5 ?
- Negative
- Positive
Answer and solution
The answer: Positive
The interval has no root and no point where the denominator is 0. Test one point in it, x = −7 : f(−7) = (−7 + 5) / (−7 − 6) = 2/13 . The value is positive, so the function is positive throughout.
A tangent to a rational graph
For example: What is the gradient of the tangent to the graph of f(x) = −12 / x at the point where x = −2 ?
Answer and solution
The answer: 3
First f′(x) = 12 / x² . Put it in: f′(−2) = 12 / (−2)² = 3 .
Reading the graphs
For example: Which graph is the graph of f(x) = 1 / (x − 3) − 2 ? Graph A:
Graph B: Graph C:- Graph A
- Graph C
- Graph B
Answer and solution
The answer: Graph B
The asymptotes are x = 3 and y = −2 . The numerator is positive, so to the right of the vertical asymptote the graph is above the horizontal one. That is graph B.
Not on the site yet
Parts of this topic that are not built yet: Sketching the graph · Complicated algebraic techniques.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 5 units (grades 10 to 12).