# Rational functions · Grade 10 · 5 units

> Rational functions, Grade 10 · 5 units: The domain and a missing point and more. Explanation, worked examples and free practice following the Israeli curriculum.

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

- Example: Which value of x is not in the domain of the function f(x) = −7 / (x − 6) ?
- Answer: 6
- Solution: The denominator x − 6 is 0 when x = 6 .

### Asymptotes

- Example: Find the vertical asymptote of the graph of f(x) = −5 / (x − 6) − 5 .
- Answer: 6
- Solution: 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

- Example: Here is the function f(x) = −5 / x . Find f′(4) .
- Answer: 5/16
- Solution: By the derivative of one over x, f′(x) = 5 / x² . Put it in: f′(4) = 5 / 4² = 5/16 .

### The quotient rule

- Example: Here is the function f(x) = (3x + 3) / (x − 2) . Find f′(0) .
- Answer: −9/4
- Solution: 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

- Example: What is the sign of the function f(x) = (x + 5) / (x − 6) for x < −5 ?
- Options: Negative · Positive
- Answer: Positive
- Solution: 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

- Example: What is the gradient of the tangent to the graph of f(x) = −12 / x at the point where x = −2 ?
- Answer: 3
- Solution: First f′(x) = 12 / x² . Put it in: f′(−2) = 12 / (−2)² = 3 .

### Reading the graphs

- Example: Which graph is the graph of f(x) = 1 / (x − 3) − 2 ? Graph A: (Graph A: a vertical asymptote at x = −3, a horizontal asymptote at y = −2, and to the right of the vertical asymptote the graph is above the horizontal one.) Graph B: (Graph B: a vertical asymptote at x = 3, a horizontal asymptote at y = −2, and to the right of the vertical asymptote the graph is above the horizontal one.) Graph C: (Graph C: a vertical asymptote at x = 3, a horizontal asymptote at y = −2, and to the right of the vertical asymptote the graph is below the horizontal one.)
- Options: Graph A · Graph C · Graph B
- Answer: Graph B
- Solution: 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

- Sketching the graph
- Complicated algebraic techniques

The exercises are made by rules and the answers are worked out in code. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 5 units (grades 10 to 12).

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