Question 1a

Marks: 1

Let $f (x) = \frac{b}{x - a} + 2$ , for $x \neq a .$

The line $x = - 3$ is a vertical asymptote of the graph of $f$ .

(a)

Write down the value of

a

.

Key Concepts

Linear Rational Functions & Graphs

Question 1b

Marks: 3

The graph of $f$ passes through the point $A (4, - 2)$ .

(b)

Find the value of

b

.

Key Concepts

Graphing Functions

Question 1c

Marks: 2

(c)

Find

\lim_{x \to \infty} f (x)

.

Key Concepts

Linear Rational Functions & Graphs

Question 2a

Marks: 6

Consider $f (x) = \frac{a x + b}{c x + 1}$ , where $a, b, c \in ℤ$ .

$y = 4$ and $x = - 1$ are the equations of the asymptotes of the graph of $f$ . Point $P (5, \frac{9}{2})$ lies on the graph.

(a)

Find the values of

a, b

and

c

.

Key Concepts

Linear Rational Functions & Graphs

Question 2b

Marks: 4

(b)

Sketch the graph of

y = f (x)

.

Key Concepts

Linear Rational Functions & Graphs
Graphing Functions

Question 3a

Marks: 1

Let $f (x) = \frac{3}{x - a} + b$ , for $x \neq a .$

The line $x = 2$ is a vertical asymptote of the graph of $f$ .

(a)

Write down the value of

a .

Key Concepts

Linear Rational Functions & Graphs

Question 3b

Marks: 3

The graph of $f$ passes through the point $A (1, 5)$ .

(b)

Find the value of

b

.

Key Concepts

Key Features of Graphs

Question 3c

Marks: 2

(c)

Find

\lim_{x \to \infty} f (x) .

Key Concepts

Limits
Linear Rational Functions & Graphs

Question 4a

Marks: 2

Consider the function $f$ defined by $f (x) = \frac{3 x - 7}{x^{2} - 6 x + 5}$ , for $x \neq 1.5 .$

(a)

Find the coordinates where the graph of

f

intersects the coordinate axes.

Key Concepts

Graphing Functions
Quadratic Rational Functions & Graphs

Question 4b

Marks: 3

(b)

(i)

Express

f (x)

as partial fractions.

(ii)

Hence, write down the equation of the horizontal asymptote.

Key Concepts

Partial Fractions

Question 4c

Marks: 4

(c)

Sketch the graph of

y = f (x)

.

Key Concepts

Quadratic Rational Functions & Graphs

Question 5a

Marks: 6

The graph of a function $f$ is shown below. The equations of the asymptotes are $x = 0.25$ and $y = 0.5 .$ The graph crosses the coordinate axes at the points $A (0, m)$ and $B (5, 0)$ .

q5a-2-5-ib-aa-hl-reciprocal-_-rational-functions-hard-maths-diagram

(a)

Find an equation for

f (x)

in the form

\frac{a x + b}{c x + d}

, where

a, b, c, d \in ℤ

.

Key Concepts

Key Features of Graphs
Linear Rational Functions & Graphs

Question 5b

Marks: 1

(b)

Write down the value of

m

.

Key Concepts

Linear Rational Functions & Graphs
Graphing Functions

Question 6a

Marks: 3

Consider the function $f (x) = \frac{2 x^{2} - x - 15}{x - 2}, x \in ℝ, x \neq 2 .$

(a)

Find the coordinates where the graph of

f

crosses the

(i)

x

-axis,

(ii)

y

-axis.

Key Concepts

Quadratic Rational Functions & Graphs
Key Features of Graphs

Question 6b

Marks: 4

(b)

Find the equation of the oblique asymptote of the graph of

f

, giving the answer in the form

y = a x + b

where

a, b \in ℤ .

.

Key Concepts

Quadratic Rational Functions & Graphs
Polynomial Division

Question 6c

Marks: 4

(c)

Sketch the graph of

y = f (x)

. Clearly indicate the asymptotes and give coordinates of the points where the graph intersects the axes.

Key Concepts

Graphing Functions
Quadratic Rational Functions & Graphs

Question 7

Marks: 6

Consider the function $f$ defined by $f (x) = \frac{a x + b}{2 x^{2} + c x + d},$ where $a, b, c, d \in ℤ$ . The lines $x = - \frac{1}{2}$ and $x = 3$ are vertical asymptotes of the graph of $f$ as shown below. The graph crosses through the points $A (0, 4)$ and $B (2, 0)$ .

q7-2-5-ib-aa-hl-reciprocal-_-rational-functions-hard-maths-diagram

Find the values of $a, b, c$ and $d$ .

Key Concepts

Quadratic Rational Functions & Graphs
Key Features of Graphs

Question 8a

Marks: 2

The graph of a function $f$ is shown below. The equations of the asymptotes are $x = 3$ and $y = 3 x - 2$ . The graph crosses the $y$ -axis at $(0, - 2.5)$ .

q8a-2-5-ib-aa-hl-reciprocal-_-rational-functions-hard-maths-diagram

The function can be written as $f (x) = a x + b + \frac{3}{c x + d}$ , where $a, b, c, d \in ℤ .$

(a)

Find the values of

a, b, c

and

d

.

Key Concepts

Quadratic Rational Functions & Graphs
Key Features of Graphs

Question 8b

Marks: 3

(b)

Find the exact coordinates of the points where

y = f (x)

crosses the

x

-axis.

Key Concepts

Solving Quadratic Equations
Key Features of Graphs

Question 8c

Marks: 2

(c)

Given that

f (x) = k

has no real solutions where

k \in ℝ

, find the set of possible values of

k

. Give the bounds correct to 2 decimal places.

Key Concepts

Intersecting Graphs

Question 9a

Marks: 4

Consider the function $f (x) = \frac{2 x^{2} - 18}{x^{2} - 25}$ , for $x \neq \pm 5$ .

(a)

(i)

Show that

\frac{2 x^{2} - 18}{x^{2} - 25} = 2 + \frac{16}{5 (x - 5)} - \frac{16}{5 (x + 5)}

.

(ii)

Hence, write down the value of

\lim_{x \to \infty} f (x)

.

Key Concepts

Quadratic Functions & Graphs

Question 9b

Marks: 4

(b)

(i)

Sketch the graph of

y = f (x)

.

(ii)

Write down the range of

f

.

Key Concepts

Graphing Functions

Question 10a

Marks: 2

Consider the function $f (x) = \frac{2 x^{2} - 72}{x^{2} - 9}$ , for $x \neq \pm 3$ .

(a)

Prove that

f

is an even function.

Key Concepts

Odd & Even Functions

Question 10b

Marks: 5

(b)

(i)

Sketch the graph of

y = f (x)

.

(ii)

Write down the range of

f

.

Key Concepts

Graphing Functions
Language of Functions

DP IB Maths: AA HL

Topic Questions

2.5 Reciprocal & Rational Functions

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