2.5 Reciprocal & Rational Functions

1 2a 2b 2c 3a 3b 3c 4a 4b 5a 5b 6a 6b 7a 7b 8a 8b 8c 9a 9b

Question 1

Marks: 7

The graph of a function $f$ is shown below. The graph crosses the $x$ -axis at the point A and the $y$ -axis at point B. The asymptotes intersect at the point C $(4, - \frac{8}{3})$ . The area of the triangle $△ AOB$ is 3.

q1-2-5-ib-aa-hl-reciprocal-_-rational-functions-very-hard-maths-diagram

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

Linear Rational Functions & Graphs
Key Features of Graphs

Question 2a

Marks: 1

Let the function $f$ be defined by $f (x) = 3 - \frac{a}{2 x - 5}, x \in ℝ, x \neq p$ , where $a$ is a positive constant.

(a)

Write down the value of

p

Key Concepts

Linear Rational Functions & Graphs
Key Features of Graphs

Question 2b

Marks: 4

(b)

Sketch the graph of

f

. State the equations of the asymptotes and give the coordinates of the intersections with the coordinate axes in terms of

a

Key Concepts

Linear Rational Functions & Graphs
Graphing Functions

Question 2c

Marks: 4

(c)

Given that the graph of

f

intersects the line given by

y = x

, find the set of possible values of a.

Key Concepts

Intersecting Graphs
Discriminants

Question 3a

Marks: 3

Consider the function defined by $f (x) = \frac{4 - 2 x}{x^{2} - 3 x + 3} .$ The graph of $f$ is shown below. The graph has a maximum point at A and a minimum point at B.

q3a-2-5-ib-aa-hl-reciprocal-_-rational-functions-very-hard-maths-diagram

(a)

Show that

f

is defined for all

x \in ℝ

Key Concepts

Quadratic Rational Functions & Graphs
Discriminants

Question 3b

Marks: 6

(b)

(i)

Given that the graph of

f

intersects the line

y = k

, show that

3 k^{2} - 4 k - 4 \leq 0

(ii)

Hence, find the range of

f

Key Concepts

Quadratic Rational Functions & Graphs
Discriminants

Question 3c

Marks: 4

(c)

Hence, find the coordinates of A and B.

Key Concepts

Solving Quadratic Equations

Question 4a

Marks: 5

Consider the function $f$ defined by $f (x) = \frac{(x - p) (x - q)}{x - n}$ , where $n, p$ and $q$ are positive constants and $p < q$ .

(a)

In the case that

p, q

and

n

are distinct:

(i)

write down the coordinates of the points where the graph of $f$ intersects the axes,

(ii)

write down the equation of the vertical asymptote,

(iii)

show that the equation of the oblique asymptote is

y = x + (n - p - q)

Key Concepts

Quadratic Rational Functions & Graphs
Key Features of Graphs

Question 4b

Marks: 6

(b)

Sketch the graph of

f

in that case that:

(i)

n < p < q,

(ii)

p < n < q

(iii)

n = p .

Clearly indicate where the graph crosses the coordinate axes and any asymptotes or discontinuities.

Key Concepts

Quadratic Rational Functions & Graphs

Question 5a

Marks: 7

(a)

Sketch the graph of the function

f

be defined by

f (x) = \frac{4 x - 1}{2 x^{2} - x - 6} .

Clearly indicate the coordinates where the graph intersects the axes and state the equation of any asymptotes.

Key Concepts

Quadratic Rational Functions & Graphs
Key Features of Graphs

Question 5b

Marks: 6

(b)

Sketch the graph of the function $g$ defined by

g

(x) = \frac{2 x^{2} - x - 6}{4 x - 1}

Clearly indicate the coordinates where the graph intersects the axes and state the equation of any asymptotes.

Key Concepts

Polynomial Division
Quadratic Rational Functions & Graphs

Question 6a

Marks: 6

The graph of a function $f$ is shown below. The graph crosses the $x$ -axis at the points $A (- 0.5, 0)$ and $B (3, 0)$ . The graph crosses the $y$ -axis at $C (0, - 3)$ . The equation of the vertical asymptote is $x = 2$ .

q6a-2-5-ib-aa-hl-reciprocal-_-rational-functions-very-hard-maths-diagram

The function $f$ can be written in the form $f (x) = \frac{a x^{2} + b x + c}{d x + e}$ , where $a, b, c, d, e \in ℤ .$

(a)

Given that

c e + 2 = b

, find the values of

a, b, c, d

and

e

Key Concepts

Quadratic Rational Functions & Graphs
Key Features of Graphs

Question 6b

Marks: 4

(b)

Hence find the equation of the oblique asymptote.

Key Concepts

Quadratic Rational Functions & Graphs
Polynomial Division

Question 7a

Marks: 8

Consider the function $f$ defined by $f (x) = \frac{a x + b}{x^{2} + c x + d}$ , where $a, b, c, d \in ℤ$ . The line $x = 2$ is the only vertical asymptote of the graph of $f$ as shown below. The graph crosses through the points $A (0, k)$ and $B (k, 0)$ where $k$ is a positive constant. The line $y = - 1$ is a tangent to the graph of $f$ at the point $C$ .

q7a-2-5-ib-aa-hl-reciprocal-_-rational-functions-very-hard-maths-diagram

(a)

Find the values of

a, b, c, d

and

k

Key Concepts

Quadratic Rational Functions & Graphs
Key Features of Graphs

Question 7b

Marks: 2

(b)

Find the coordinates of

C

Key Concepts

Intersecting Graphs
Solving Quadratic Equations

Question 8a

Marks: 2

Consider the function $f (x) = \frac{2 x^{2} + 5 x - 12}{x^{2} - 4}, x \in ℝ, x \neq \pm 2 .$

Solve

f (x) = 0

Key Concepts

Solving Quadratic Equations

Question 8b

Marks: 3

(b)

Show that

\frac{2 x^{2} + 5 x - 12}{x^{2} - 4} = A + \frac{B x + C}{x^{2} - 4}

, where

A, B

and

C

are constants to be found.

Key Concepts

Polynomial Division

Question 8c

Marks: 6

(c)

Using the answer to part (b), sketch the graph of

f

. Clearly indicate the coordinates of the points where the graph intersects the axes and state the equations of any asymptotes.

Key Concepts

Quadratic Rational Functions & Graphs
Key Features of Graphs

Question 9a

Marks: 3

The diagram below shows the graphs of two linear functions $f$ and $g$ and a quadratic function $h$ .

q9a-2-5-ib-aa-hl-reciprocal-_-rational-functions-very-hard-maths-diagram

(a)

Sketch the graph of

y = \frac{f (x)}{g (x)}

. Clearly indicate where the graph intersects the

x

-axis and the location of any vertical asymptotes.

Key Concepts

Linear Rational Functions & Graphs

Question 9b

Marks: 4

(b)

Sketch the graph of

y = \frac{g (x)}{h (x)}

. Clearly indicate where the graph intersects the

x

-axis and the location of any vertical asymptotes.

Key Concepts

Quadratic Rational Functions & Graphs

DP IB Maths: AA HL

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2.5 Reciprocal & Rational Functions

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