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Reciprocal & Rational Functions (DP IB Analysis & Approaches (AA))

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FrontReciprocal & Rational Functions
What does the graph of the function $f (x) = \frac{1}{x}, x \neq 0$ look like?
FrontReciprocal & Rational Functions
What are the equations of the asymptotes of the function $f (x) = \frac{1}{x}, x \neq 0$ ?
BackReciprocal & Rational Functions
The equations of the asymptotes of the function $f (x) = \frac{1}{x}, x \neq 0$ are:
$x = 0$
$y = 0$

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What does the graph of the function $f (x) = \frac{1}{x}, x \neq 0$ look like?
The graph of $f (x) = \frac{1}{x}, x \neq 0$ is shown below.
True or False?
The inverse of the reciprocal function, $f (x) = \frac{1}{x}$ , is itself.
True.
The inverse of the reciprocal function, $f (x) = \frac{1}{x}$ , is itself.
The reciprocal function is a self-inverse function.
What are the equations of the asymptotes of the function $f (x) = \frac{1}{x}, x \neq 0$ ?
The equations of the asymptotes of the function $f (x) = \frac{1}{x}, x \neq 0$ are:
- $x = 0$
- $y = 0$
What is the range of the reciprocal function, $f (x) = \frac{1}{x}, x \neq 0$ ?
The range of the reciprocal function, $f (x) = \frac{1}{x}, x \neq 0$ is all real number values except zero: $f (x) \in ℝ, f (x) \neq 0$ .
How do you find the equation of the vertical asymptote of the rational function $f (x) = \frac{a x + b}{c x + d}$ ?
To find the equation of the vertical asymptote of the rational function $f (x) = \frac{a x + b}{c x + d}$ , you set the denominator equal to zero.
The equation is $c x + d = 0$ which can be written as $x = - \frac{d}{c}$ .
How do you find the equation of the horizontal asymptote of the rational function $f (x) = \frac{a x + b}{c x + d}$ ?
To find the equation of the horizontal asymptote of the rational function $f (x) = \frac{a x + b}{c x + d}$ , you look at the coefficients of the $x$ terms.
The equation is $y = \frac{a}{c}$ .
True or False?
Any rational function of the form $f (x) = \frac{a x + b}{c x + d}$ , will always have exactly one real root.
False.
If $a = 0$ , the rational function will not have any real roots as the horizontal asymptote is $y = 0$ .
Any rational function of the form $f (x) = \frac{a x + b}{c x + d}$ , will always have exactly one real root provided $a \neq 0$ .
If you are asked to sketch a rational function of the form, $f (x) = \frac{a x + b}{c x + d}$ , what should you include in the sketch?
If you are asked to sketch a rational function of the form, $f (x) = \frac{a x + b}{c x + d}$ , you should include:
- the horizontal asymptote and its equation,
- the vertical asymptote and its equation,
- the coordinates of any intercepts with the axes.
True or False?
The inverse of any rational function of the form $f (x) = \frac{a x + b}{c x + d}$ is itself.
False.
The inverse of any rational function of the form $f (x) = \frac{a x + b}{c x + d}$ is not always itself.
How many real roots can the graph of the rational function $f (x) = \frac{a x^{2} + b x + c}{d x + e}$ have?
The graph of the rational function $f (x) = \frac{a x^{2} + b x + c}{d x + e}$ can have 0, 1 or 2 real roots. They will be solutions to the equation $a x^{2} + b x + c = 0$ .
How do you find the equation of the vertical asymptote of the rational function $f (x) = \frac{a x^{2} + b x + c}{d x + e}$ ?
To find the equation of the vertical asymptote of the rational function $f (x) = \frac{a x^{2} + b x + c}{d x + e}$ , you set the denominator equal to zero.
$d x + e = 0$ can be written as $x = - \frac{e}{d}$ .
What is an oblique asymptote of a graph?
An oblique asymptote is a slanted (not horizontal nor vertical) straight line that the graph tends towards as $x \to \infty$ and/or as $x \to - \infty$
True or False?
The rational function $f (x) = \frac{a x^{2} + b x + c}{d x + e}$ has a horizontal asymptote.
False.
The rational function $f (x) = \frac{a x^{2} + b x + c}{d x + e}$ does not have a horizontal asymptote. It is an oblique asymptote.
(It also has a vertical asymptote at $x = - \frac{e}{d}$ .)
How do you find the equation of the oblique asymptote of the function $f (x) = \frac{a x^{2} + b x + c}{d x + e}$ ?
To find the equation of the oblique asymptote of the function $f (x) = \frac{a x^{2} + b x + c}{d x + e}$ , you write $a x^{2} + b x + c$ in the form $(d x + e) (p x + q) + r$ . This can be done by using polynomial division or by comparing coefficients.
The equation of the oblique asymptote is then $y = p x + q$ .
True or False?
The graph of the rational function $f (x) = \frac{a x + b}{c x^{2} + d x + e}$ always has one real root (provided $a \neq 0$ , and $a x + b$ is not a factor of $c x^{2} + d x + e$ ).
True.
The graph of the rational function $f (x) = \frac{a x + b}{c x^{2} + d x + e}$ always has one real root (provided $a \neq 0$ , and $a x + b$ is not a factor of $c x^{2} + d x + e$ ).
The root is the solution to the equation $a x + b = 0$ .
What is the equation of the horizontal asymptote of the graph of the rational function $f (x) = \frac{a x + b}{c x^{2} + d x + e}$ ?
The equation of the horizontal asymptote of the graph of the rational function $f (x) = \frac{a x + b}{c x^{2} + d x + e}$ is always $y = 0$ , i.e. the $x$ -axis.
True or False?
The graph of the rational function $f (x) = \frac{a x + b}{c x^{2} + d x + e}$ always has exactly two vertical asymptotes.
False.
The graph of the rational function $f (x) = \frac{a x + b}{c x^{2} + d x + e}$ can have 0, 1 or 2 vertical asymptotes.
The number of asymptotes is the same as the number of real solutions to $c x^{2} + d x + e = 0$ .
If you draw a horizontal line, what is the maximum number of times it can intersect the graph of a rational function with an equation of the form $f (x) = \frac{a x^{2} + b x + c}{d x + e}$ or $f (x) = \frac{a x + b}{c x^{2} + d x + e}$ ?
If you draw a horizontal line, it can intersect the graph of a rational function with an equation of the form $f (x) = \frac{a x^{2} + b x + c}{d x + e}$ or $f (x) = \frac{a x + b}{c x^{2} + d x + e}$ at most two times.

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Reciprocal & Rational Functions (DP IB Analysis & Approaches (AA))

Flashcards

Cards in this collection (18)

1. Number & Algebra

Number & Algebra Toolkit

Exponentials & Logs

Sequences & Series

Simple Proof & Reasoning

Proof by Induction & Contradiction

Binomial Theorem

Permutations & Combinations

Complex Numbers

Further Complex Numbers

Systems of Linear Equations

2. Functions

Linear Functions & Graphs

Quadratic Functions & Graphs

Functions Toolkit

Other Functions & Graphs

Reciprocal & Rational Functions

Transformations of Graphs

Polynomial Functions

Inequalities

Modulus Functions & Further Transformations

3. Geometry & Trigonometry

Geometry Toolkit

Geometry of 3D Shapes

Trigonometry

The Unit Circle & Exact Values

Trigonometric Functions & Graphs

Trigonometric Equations & Identities

Inverse & Reciprocal Trigonometric Functions

Trigonometric Proof & Equation Strategies

Vector Properties

Vector Equations of Lines

Vector Planes

4. Statistics & Probability

Statistics Toolkit

Correlation & Regression

Probability

Discrete Random Variables

Binomial Distribution

Normal Distribution

Continuous Random Variables

5. Calculus

Differentiation

Techniques & Applications of Differentiation

Integration

Techniques & Applications of Integration

Optimisation

Kinematics

Basic Limits & Continuity

Further Differentiation

Further Integration

Differential Equations

Maclaurin Series

Limits using l'Hôpital's Rule & Maclaurin Series