Reciprocal & Rational Functions (DP IB Maths: AA HL)

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Dan

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Reciprocal Functions & Graphs

What is the reciprocal function?

The reciprocal function is defined by $f (x) = \frac{1}{x}, x \neq 0$
Its domain is the set of all real values except 0
Its range is the set of all real values except 0
The reciprocal function has a self-inverse nature
- $f^{- 1} (x) = f (x)$
- $(f \circ f) (x) = x$

What are the key features of the reciprocal graph?

The graph does not have a y-intercept
The graph does not have any roots
The graph has two asymptotes
- A horizontal asymptote at the x-axis: $y = 0$
  - This is the limiting value when the absolute value of x gets very large
- A vertical asymptote at the y-axis: $x = 0$
  - This is the value that causes the denominator to be zero
The graph has two axes of symmetry
- $y = x$
- $y = - x$
The graph does not have any minimum or maximum points

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Linear Rational Functions & Graphs

What is a rational function with linear terms?

A (linear) rational function is of the form $f (x) = \frac{a x + b}{c x + d}, x \neq - \frac{d}{c}$
Its domain is the set of all real values except $- \frac{d}{c}$
Its range is the set of all real values except $\frac{a}{c}$
The reciprocal function is a special case of a rational function

What are the key features of linear rational graphs?

The graph has a y-intercept at $(0, \frac{b}{d})$ provided $d \neq 0$
The graph has one root at $(- \frac{b}{a}, 0)$ provided $a \neq 0$
The graph has two asymptotes
- A horizontal asymptote: $y = \frac{a}{c}$
  - This is the limiting value when the absolute value of x gets very large
- A vertical asymptote: $x = - \frac{d}{c}$
  - This is the value that causes the denominator to be zero
The graph does not have any minimum or maximum points
If you are asked to sketch or draw a rational graph:
- Give the coordinates of any intercepts with the axes
- Give the equations of the asymptotes

Exam Tip

If you draw a horizontal line anywhere it should only intersect this type of graph once at most
The only horizontal line that should not intersect the graph is the horizontal asymptote
- This can be used to check your sketch in an exam

Worked example

The function $f$ is defined by $f (x) = \frac{10 - 5 x}{x + 2}$ for $x \neq - 2$ .

Write down the equation of

(i)

the vertical asymptote of the graph of

f

(ii)

the horizontal asymptote of the graph of

f

2-4-1-ib-aa-sl-rational-func-a-we-solution

Find the coordinates of the intercepts of the graph of

f

with the axes.

2-4-1-ib-aa-sl-rational-func-b-we-solution

Sketch the graph of

f

2-4-1-ib-aa-sl-rational-func-c-we-solution

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Quadratic Rational Functions & Graphs

How do I sketch the graph of a rational function where the terms are not linear?

A rational function can be written $f (x) = \frac{g (x)}{h (x)}$

Where g and h are polynomials

To find the y-intercept evaluate $\frac{g (0)}{h (0)}$
To find the x-intercept(s) solve $g (x) = 0$
To find the equations of the vertical asymptote(s) solve $h (x) = 0$
There will also be an asymptote determined by what f(x) tends to as x approaches infinity

In this course it will be either:

Horizontal
Oblique (a slanted line)

This can be found by writing $g (x)$ in the form $h (x) Q (x) + r (x)$

You can do this by polynomial division or comparing coefficients

The function then tends to the curve $y = Q (x)$

What are the key features of rational graphs: quadratic over linear?

For the rational function of the form $f (x) = \frac{a x^{2} + b x + c}{d x + e}$
The graph has a y-intercept at $(0, \frac{c}{e})$ provided $e \neq 0$
The graph can have 0, 1 or 2 roots

They are the solutions to $a x^{2} + b x + c = 0$

The graph has one vertical asymptote $x = - \frac{e}{d}$
The graph has an oblique asymptote $y = p x + q$

Which can be found by writing $a x^{2} + b x + c$ in the form $(d x + e) (p x + q) + r$

Where p, q, r are constants
This can be done by polynomial division or comparing coefficients

What are the key features of rational graphs: linear over quadratic?

For the rational function of the form $f (x) = \frac{a x + b}{c x^{2} + d x + e}$
The graph has a y-intercept at $(0, \frac{b}{e})$ provided $e \neq 0$
The graph has one root at $x = - \frac{b}{a}$
The graph has can have 0, 1 or 2 vertical asymptotes

They are the solutions to $c x^{2} + d x + e = 0$

The graph has a horizontal asymptote

Exam Tip

If you draw a horizontal line anywhere it should only intersect this type of graph twice at most
- This idea can be used to check your graph or help you sketch it

Worked example

The function $f$ is defined by $f (x) = \frac{2 x^{2} + 5 x - 3}{x + 1}$ for $x \neq - 1$ .

(i)

Show that

\frac{2 x^{2} + 5 x - 3}{x + 1} = p x + q + \frac{r}{x + 1}

for constants

p, q

and

r

which are to be found.

(ii)

Hence write down the equation of the oblique asymptote of the graph of

f

2-5-1-ib-aa-hl-quad-rational-function-a-we-solution

Find the coordinates of the intercepts of the graph of

f

with the axes.

2-5-1-ib-aa-hl-quad-rational-function-b-we-solution

Sketch the graph of

f

2-5-1-ib-aa-hl-quad-rational-function-c-we-solution

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Reciprocal & Rational Functions (DP IB Maths: AA HL)

Revision Note

What is the reciprocal function?

What are the key features of the reciprocal graph?

What is a rational function with linear terms?

What are the key features of linear rational graphs?

How do I sketch the graph of a rational function where the terms are not linear?

What are the key features of rational graphs: quadratic over linear?

What are the key features of rational graphs: linear over quadratic?

Author: Dan