Rational Functions - IB Math AA Question Bank

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Rational Functions

Practice exam-style IB Math AA questions for Rational Functions, aligned with the syllabus and grouped by topic.

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Question 1

SL • Paper 1

Easy

Non Calculator

SL • Paper 1

Easy

Non Calculator

The function $f$ is defined by $f(x)=\dfrac{ax+b}{x-2}$ , for $x\neq 2$ . The graph of $y=f(x)$ has horizontal asymptote $y=4$ and $x$ -intercept $(-1,0)$ .

Find the values of $a$ and $b$ .

[3]

Find the $y$ -intercept of the graph.

[1]

Question 2

SL • Paper 2

Easy

Calculator Permitted

SL • Paper 2

Easy

Calculator Permitted

Consider $g(x)=\dfrac{kx+1}{x-4}$ , where $x\ne 4$ . The graph of $y=g(x)$ has horizontal asymptote $y=3$ .

Find the value of $k$ .

[1]

Find the coordinates of the intercepts of the graph with the coordinate axes.

[2]

Solve $g(x)=5.5$ .

[2]

Question 3

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The function $f$ is defined by $f(x)=\dfrac{3x-2}{x+4}$ , for $x\neq -4$ .

Write down the equations of the vertical and horizontal asymptotes of the graph of $y=f(x)$ .

[2]

Find the coordinates of the intercepts of the graph with the axes.

[2]

Solve $f(x)=1$ .

[2]

Question 4

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Let $f(x)=a+\dfrac{b}{x-c}$ , for $x\neq c$ . The graph of $y=f(x)$ has asymptotes $x=3$ and $y=-2$ , and passes through the point $P(1,5)$ .

Write down the values of $a$ and $c$ .

[2]

Find the value of $b$ .

[2]

Find the $x$ -intercept of the graph of $y=f(x)$ .

[1]

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Question 5

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The function $f$ is defined by $f(x)=\dfrac{2x+6}{x-1}$ , for $x\neq 1$ .

Show that $f(x)=2+\dfrac{8}{x-1}$ .

[2]

Describe fully the transformations that map the graph of $y=\dfrac{1}{x}$ to the graph of $y=f(x)$ .

[2]

State the domain and range of $f$ .

[2]

Question 6

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Let $f(x)=\dfrac{kx+3}{2x-k}$ , where $k$ is a constant. The graph of $y=f(x)$ has vertical asymptote $x=2$ .

Find the value of $k$ .

[2]

Write down the equation of the horizontal asymptote.

[1]

Find the coordinates of the intercepts of the graph with the axes.

[2]

Question 7

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Consider the rational expression $\dfrac{7x+1}{(x-2)(x+1)}$ .

Find constants $A$ and $B$ such that $\dfrac{7x+1}{(x-2)(x+1)}\equiv \dfrac{A}{x-2}+\dfrac{B}{x+1}$ .

[3]

Hence solve $\dfrac{7x+1}{(x-2)(x+1)}=3$ .

[3]

Question 8

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

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

State the domain of $f$ and the equation of the vertical asymptote.

[2]

Find the equation of the oblique asymptote.

[2]

Find the $y$ -intercept and justify why the graph has no $x$ -intercepts.

[2]

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Question 9

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The function $f$ is defined by $f(x)=\dfrac{2x-1}{x^2-x-6}$ .

Find the equations of all vertical asymptotes of the graph of $y=f(x)$ .

[2]

Write down the equation of the horizontal asymptote.

[1]

Find the coordinates of the intercepts of the graph with the axes.

[2]

Question 10

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

Let $f(x)=\dfrac{3x-7}{2x+5}$ , for $x\ne -\dfrac{5}{2}$ .

Write down the equations of the vertical and horizontal asymptotes of the graph of $y=f(x)$ .

[2]

Find the coordinates of the intercepts of the graph with the coordinate axes.

[2]

Find the coordinates of the points of intersection of $y=f(x)$ and $y=-x$ .

[2]

Question 11

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A function $f$ has the form $f(x)=\dfrac{a}{x-h}+k$ , where $a$ , $h$ and $k$ are constants. The graph of $y=f(x)$ has asymptotes $x=3$ and $y=-2$ , and passes through the point $(5,4)$ .

Determine the values of $a$ , $h$ and $k$ .

[2]

Find the $x$ -intercept of the graph of $y=f(x)$ .

[1]

Find the coordinates of the points where the graph of $y=f(x)$ intersects the line $y=x$ .

[2]

Question 12

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The resistance $R$ (in ohms) of a component at time $t$ (in seconds) is modelled by

R(t)=\frac{at+b}{t+c}, \quad t\ge 0

where $a$ , $b$ and $c$ are constants. The rational function, considered beyond the physical domain $t\ge 0$ , has vertical asymptote $t=-2\ \text{s}$ , horizontal asymptote $R=12\ \Omega$ , and $R(0)=5\ \Omega$ .

Determine the values of $a$ , $b$ and $c$ .

[3]

Find the value of $t$ for which $R(t)=9$ .

[2]

State whether the value of $t$ found in part (b) is valid for this model.

[1]

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Question 13

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

Let $f(x)=\dfrac{x+1}{x-2}$ , for $x\ne 2$ .

Write down the equations of the asymptotes of the graph of $y=f(x)$ .

[2]

Find the coordinates of the intercepts of the graph with the coordinate axes.

[2]

Hence sketch the graph of $y=f(x)$ , clearly showing the asymptotes and intercepts.

[2]

Question 14

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The function $f$ is defined by $f(x)=\dfrac{ax+b}{x+d}$ , where $x\ne -d$ . The graph of $y=f(x)$ has vertical asymptote $x=-3$ , horizontal asymptote $y=2$ , and $x$ -intercept $\left(\dfrac{3}{2},0\right)$ .

Determine the values of $a$ , $b$ and $d$ .

[3]

Write $f(x)$ using the values found in part (a).

[1]

Find the $y$ -intercept of the graph of $y=f(x)$ .

[1]

Question 15

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let $f(x)=\dfrac{2x+1}{x^2-5x+6}$ .

Find the domain of $f$ and the equations of any vertical asymptotes.

[2]

Find the horizontal asymptote and the $x$ -intercept of the graph of $y=f(x)$ .

[2]

Using a GDC, find the coordinates of the points where $y=f(x)$ intersects the line $y=0.7$ .

[2]

Question 16

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let

h(x)=\frac{4x-1}{(x-1)(x+2)}, \quad x\ne 1,-2

Express $h(x)$ in partial fractions.

[3]

Using a GDC, solve $h(x)=1.2$ .

[2]

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Question 17

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The function $h$ is defined by $h(x)=\dfrac{1}{x-2}+3$ , for $x\neq 2$ .

State the domain and range of $h$ .

[2]

Find the coordinates of the intercepts of the graph of $y=h(x)$ with the axes.

[2]

Solve $h(x)=x$ .

[2]

Question 18

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $n$ be a positive integer.

Show that $\dfrac{1}{(r+1)(r+3)}\equiv \dfrac12\left(\dfrac{1}{r+1}-\dfrac{1}{r+3}\right)$ .

[2]

Hence find $\displaystyle \sum_{r=1}^{n}\dfrac{1}{(r+1)(r+3)}$ in terms of $n$ .

[3]

Question 19

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The function $f$ is defined by $f(x)=\dfrac{x+a}{x^2+bx+4}$ . The graph of $y=f(x)$ has vertical asymptotes $x=1$ and $x=4$ , and passes through the point $(0,-\dfrac12)$ .

Find the value of $b$ .

[2]

Find the value of $a$ .

[2]

State the domain of $f$ and find the $x$ -intercept of the graph.

[2]

Question 20

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The function $f$ is defined by $f(x)=\dfrac{2x^2+px+q}{x-1}$ , for $x\neq 1$ . The graph of $y=f(x)$ has oblique asymptote $y=2x+3$ and $x$ -intercept $(2,0)$ .

Find the value of $p$ .

[2]

Find the value of $q$ .

[2]

Find the equation of the vertical asymptote and the $y$ -intercept of the graph.

[2]

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Question 21

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

For positive integers $n$ , define

S_n=\sum_{r=1}^{n}\frac{10}{(r+1)(r+3)}

Express $\dfrac{10}{(r+1)(r+3)}$ in partial fractions.

[2]

Hence show that $S_n=\dfrac{25}{6}-\dfrac{5}{n+2}-\dfrac{5}{n+3}$ .

[2]

Find the least value of $n$ for which $S_n>4$ .

[2]

Question 22

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Consider the rational function

f(x)=\frac{x^2-4x+1}{x-2}, \quad x\ne 2

Show that $f(x)=x-2-\dfrac{3}{x-2}$ .

[2]

Write down the equations of the vertical and oblique asymptotes of the graph of $y=f(x)$ .

[2]

Find the $x$ -intercepts of the graph of $y=f(x)$ .

[1]

Find the coordinates of the points where the graph of $y=f(x)$ intersects the line $y=4$ .

[2]

Question 23

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

For $k\in\mathbb{R}$ , let

f_k(x)=\frac{x+k}{x^2-2kx+9}

Write down the equation of the horizontal asymptote of the graph of $y=f_k(x)$ .

[1]

Determine the set of values of $k$ for which the graph of $y=f_k(x)$ has no vertical asymptotes.

[3]

For $k=4$ , find the equations of the vertical asymptotes of the graph of $y=f_k(x)$ .

[2]

Question 24

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let $f(x)=\dfrac{x^2+px+q}{x-1}$ , where $p$ and $q$ are constants. The graph of $y=f(x)$ has oblique asymptote $y=x+3$ and $y$ -intercept $(0,4)$ .

Determine the values of $p$ and $q$ .

[2]

Hence write $f(x)$ in the form $x+3+\dfrac{A}{x-1}$ .

[2]

Find the $x$ -intercepts of the graph of $y=f(x)$ .

[2]

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Question 25

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Let $f(x)=\dfrac{4x-5}{x+2}$ , where $x\neq -2$ .

I.

Show that $f(x)=4-\dfrac{13}{x+2}$ .

[2]

II.

Write down the equations of the asymptotes of the graph of $y=f(x)$ .

[2]

Find the coordinates of the intercepts of the graph of $y=f(x)$ with the coordinate axes.

[3]

Find the coordinates of the points of intersection of the graph of $y=f(x)$ and the line $y=-x+4$ .

[2]

Hence solve $f(x)>-x+4$ .

[3]

Question 26

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

A function has the form $f(x)=a+\dfrac{b}{x-h}$ , where $a$ , $b$ and $h$ are constants. The graph of $y=f(x)$ has asymptotes $x=-2$ and $y=3$ , and passes through the point $(0,1)$ .

I.

Write down the values of $a$ and $h$ .

[2]

II.

Find the value of $b$ .

[2]

State the domain and range of $f$ , and find the $x$ -intercept of the graph.

[4]

I.

Find the coordinates of the points where the graph of $y=f(x)$ intersects the line $y=x$ .

[3]

II.

Find the exact distance between these two points.

[1]

Question 27

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The concentration $C$ of a solution, in grams per litre, is modelled by

C(t)=8-\frac{A}{t+b}, \quad t\geq 0

where $t$ is the time in minutes and $A$ and $b$ are positive constants. It is given that $C(0)=2$ and $C(4)=5$ .

I.

Use $C(0)=2$ to form an equation involving $A$ and $b$ .

[2]

II.

Find the values of $A$ and $b$ .

[3]

Find the time at which $C(t)=6$ .

[2]

State the range of values of $C(t)$ for this model, justifying your answer.

[3]

Question 28

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The function $f$ is defined by

f(x)=\frac{5x-1}{x+2}, \quad x\ne -2

I.

Write down the equations of the vertical and horizontal asymptotes of the graph of $y=f(x)$ .

[2]

II.

Find the coordinates of the intercepts of the graph of $y=f(x)$ with the coordinate axes.

[2]

The graph of $y=f(x)$ intersects the line $y=x$ at two points $A$ and $B$ . Find the coordinates of $A$ and $B$ .

[3]

Hence find the equation of the perpendicular bisector of $AB$ .

[3]

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Question 29

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A sensor reading $R$ is modelled by

R(t)=\frac{a}{t-h}+k, \quad t>1

where $t$ is the time in seconds and $a$ , $h$ and $k$ are constants. The graph of $R$ has vertical asymptote $t=1$ and horizontal asymptote $R=20$ . It is known that $R(3)=50$ and $R(6)=32$ .

I.

Write down the values of $h$ and $k$ .

[2]

II.

Find the value of $a$ .

[2]

Find the time at which the sensor reading is $40$ .

[2]

Using the model, determine the least integer value of $t$ for which $R(t)<25$ .

[3]

Question 30

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A function has the form

f(x)=k+\frac{a}{x-2}, \quad x\ne 2

The graph of $y=f(x)$ passes through the points $(4,7)$ and $(8,4)$ .

I.

Use the two given points to form two equations in $a$ and $k$ .

[2]

II.

Hence find the values of $a$ and $k$ .

[2]

Find the coordinates of the $x$ -intercept of the graph of $y=f(x)$ .

[2]

Find the coordinates of the points where the graph of $y=f(x)$ intersects the line $y=\frac{x}{2}$ .

[3]

Write down the equations of the asymptotes of the graph of $y=f(x)$ .

[1]

Question 31

SL • Paper 1

Hard

Non Calculator

SL • Paper 1

Hard

Non Calculator

Let $f(x)=2+\dfrac{9}{x-2}$ , where $x\neq 2$ .

I.

Write down the equations of the asymptotes of the graph of $y=f(x)$ .

[2]

II.

Find the coordinates of the intercepts with the coordinate axes.

[2]

Show that $f$ is a self-inverse function.

[3]

Find the points where the graph of $y=f(x)$ intersects the line $y=x$ .

[2]

Sketch the graph of $y=f(x)$ , showing the asymptotes and the points found in part (c).

[2]

Question 32

SL • Paper 1

Hard

Non Calculator

SL • Paper 1

Hard

Non Calculator

Let $f(x)=\dfrac{ax+b}{2x-1}$ , where $x\neq \dfrac{1}{2}$ . The graph of $y=f(x)$ has horizontal asymptote $y=-3$ and $x$ -intercept $(4,0)$ .

I.

Find the value of $a$ .

[2]

II.

Find the value of $b$ .

[2]

Find the $y$ -intercept and state the vertical asymptote of the graph.

[2]

Solve $f(x)>0$ .

[3]

State the value of $k$ for which the equation $f(x)=k$ has no solution.

[1]

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Question 33

SL • Paper 1

Hard

Non Calculator

SL • Paper 1

Hard

Non Calculator

Let $f(x)=2+\dfrac{6}{x-1}$ , where $x\neq 1$ . A second function $g$ is defined by $g(x)=f(-x)$ .

I.

Find an expression for $g(x)$ .

[2]

II.

Describe the transformation that maps the graph of $y=f(x)$ to the graph of $y=g(x)$ .

[1]

Write down the vertical asymptotes of the graphs of $y=f(x)$ and $y=g(x)$ , and their common horizontal asymptote.

[3]

Find the coordinates of the point of intersection of the graphs of $y=f(x)$ and $y=g(x)$ .

[2]

Solve $f(x)>g(x)$ .

[3]

Question 34

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

For a positive integer $n$ , define

S_n=\sum_{r=1}^{n}\frac{3}{(r+2)(r+5)}

I.

Find constants $A$ and $B$ such that $\dfrac{3}{(r+2)(r+5)}\equiv\dfrac{A}{r+2}+\dfrac{B}{r+5}$ .

[3]

Hence show that

S_n=\frac{47}{60}-\frac{1}{n+3}-\frac{1}{n+4}-\frac{1}{n+5}

[4]

Find $\displaystyle\lim_{n\to\infty}S_n$ .

[2]

Show that $22$ is the least value of $n$ for which $S_n>\dfrac{2}{3}$ .

[3]

Question 35

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let

f(x)=\frac{2x+3}{x^2-4x+3}

I.

Find the domain of $f$ and the equations of the vertical asymptotes.

[2]

II.

Write down the equation of the horizontal asymptote.

[1]

Find the coordinates of the intercepts with the coordinate axes.

[3]

Express $f(x)$ in partial fractions.

[3]

Solve $f(x)\geq 1$ .

[4]

Question 36

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let

f(x)=\frac{x^2-3x+5}{x-2}, \quad x\neq 2

I.

Show that $f(x)=x-1+\dfrac{3}{x-2}$ .

[2]

II.

Write down the equations of the asymptotes of the graph of $y=f(x)$ .

[2]

Find the $y$ -intercept and show that the graph has no $x$ -intercepts.

[3]

Find the coordinates of the point where the graph of $y=f(x)$ intersects the line $y=x+2$ .

[2]

Determine the values of the constant $k$ for which the line $y=x+k$ does not intersect the graph of $y=f(x)$ .

[3]

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Question 37

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

Let

f(x)=\frac{2x+7}{x-3}, \quad x\ne 3

I.

Write down the equations of the asymptotes of the graph of $y=f(x)$ .

[2]

II.

State the range of $f$ .

[1]

Find $f^{-1}(x)$ and state its domain.

[3]

The graphs of $y=f(x)$ and $y=x$ intersect at two points. Find their coordinates, giving your answers to three significant figures.

[3]

Explain why the points found in part (c) are intersection points of the graphs of $y=f(x)$ and $y=f^{-1}(x)$ .

[3]

Question 38

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

Consider the function

g(x)=4-\frac{3}{x+1}, \quad x\ne -1

I.

Write down the equations of the asymptotes of the graph of $y=g(x)$ .

[2]

II.

Describe the transformations that map the graph of $y=\frac{1}{x}$ to the graph of $y=g(x)$ .

[1]

Find the coordinates of the intercepts of the graph of $y=g(x)$ with the coordinate axes.

[2]

Solve $g(x)>x$ . Give exact values for any boundary points.

[4]

Question 39

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

The function $f$ is defined by

f(x)=\frac{x+4}{2x-1}, \quad x\ne \frac12

I.

Find $f(f(x))$ .

[2]

II.

State what this implies about $f^{-1}(x)$ .

[1]

Find the two fixed points of $f$ .

[3]

Find the equation of the perpendicular bisector of the line segment joining the two fixed points.

[3]

Question 40

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

For $n\in\mathbb{Z}^+$ with $n\ge4$ , define

S_n=\sum_{r=1}^{n}\frac{4}{(r+2)(r+4)}

I.

Express $\frac{4}{(r+2)(r+4)}$ in partial fractions.

[2]

II.

Write out the first four terms of $S_n$ using your result from part (a)(i).

[1]

Show that

S_n=\frac76-\frac{2}{n+3}-\frac{2}{n+4}

[4]

Find the least value of $n$ for which $S_n>1.1$ .

[3]

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Question 41

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Consider the rational function

f(x)=\frac{x^2+2x+5}{x-1}, \quad x\ne 1

I.

Show that $f(x)=x+3+\frac{8}{x-1}$ .

[2]

II.

Write down the equations of the vertical and oblique asymptotes.

[1]

Find the $y$ -intercept and justify why the graph has no $x$ -intercepts.

[3]

Find the coordinates of the points where the graph of $y=f(x)$ intersects the line $y=2x$ .

[3]

Find the area between the graph of $y=f(x)$ and its oblique asymptote for $2\le x\le 6$ .

[3]

Question 42

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Let

h(x)=\frac{5x+1}{(x+1)(x+4)}, \quad x\ne -1,-4

I.

Express $h(x)$ in partial fractions.

[2]

II.

Write down the equation of the horizontal asymptote of $y=h(x)$ .

[1]

Find the equations of the vertical asymptotes and the coordinates of the intercepts with the axes.

[3]

Show that the graph of $y=h(x)$ does not intersect the line $y=1$ .

[2]

Find the area between the graph of $y=h(x)$ , the $x$ -axis, and the lines $x=0$ and $x=3$ .

[3]

Question 43

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

This question investigates a finite sum whose terms are rational functions. For $n\in\mathbb{Z}^+$ , define

S_n=\sum_{r=1}^{n}\frac{3}{(r+2)(r+5)}

I.

Show that

\frac{3}{(r+2)(r+5)}\equiv \frac{1}{r+2}-\frac{1}{r+5}

[2]

II.

Hence find $S_n$ in terms of $n$ .

[2]

I.

Find $\displaystyle \lim_{n\to\infty}S_n$ .

[1]

II.

Find the least value of $n$ for which $S_n$ is within $0.001$ of its limiting value.

[3]

III.

Explain why $S_n$ is increasing.

[2]

Question 44

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The graph of

f(x)=\frac{5x+1}{x^2-x-6}

is shown for an interval not containing its vertical asymptotes.

I.

Express $f(x)$ in partial fractions.

[3]

II.

Write down the equations of the asymptotes of the graph of $y=f(x)$ .

[1]

I.

Find the exact area between the graph of $y=f(x)$ and the $x$ -axis for $0\leq x\leq 1$ .

[3]

II.

Find the value of $a$ , $0<a<1$ , for which the area between the graph of $y=f(x)$ and the $x$ -axis for $0\leq x\leq a$ is $0.4$ .

[3]

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Question 45

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $a\in\mathbb{R}$ , consider the rational function

g_a(x)=\frac{x^2+ax+4}{x-2},\quad x\neq 2

I.

Show that

g_a(x)=x+a+2+\frac{2a+8}{x-2}

[2]

II.

State the equations of the asymptotes of the graph of $y=g_a(x)$ , for $a\neq -4$ .

[2]

For the rest of this question, take $a=-1$ .

I.

Find the coordinates of the stationary points of the graph of $y=g_{-1}(x)$ .

[3]

II.

Use algebra to verify the intercepts shown on the graph of $y=g_{-1}(x)$ .

[2]

III.

Use the supplied graph and the expression $g_{-1}(x)=x+1+\frac{6}{x-2}$ to explain why the left and right branches lie on opposite sides of the oblique asymptote.

[1]

Question 46

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $k\in\mathbb{R}$ , define

f_k(x)=\frac{x+k}{x^2-2kx+5}

This question considers how changing $k$ affects the vertical asymptotes and an area under the curve.

I.

Find the set of values of $k$ for which the graph of $y=f_k(x)$ has no vertical asymptote.

[2]

II.

State the equation of the horizontal asymptote of the graph of $y=f_k(x)$ .

[1]

III.

For $k=3$ , write $f_3(x)$ in partial fractions.

[2]

Use $f_3(x)$ from part (a).

I.

Show that $f_3(x)<0$ for $2\leq x\leq 4$ .

[2]

II.

Find the exact area between the graph of $y=f_3(x)$ and the $x$ -axis for $2\leq x\leq 4$ .

[3]

Question 47

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A rational function has the form

h(x)=\frac{ax^2+bx+c}{x-3},\quad x\neq 3

Its oblique asymptote is $y=2x-1$ and its $y$ -intercept is $(0,-2)$ .

I.

Determine $a$ , $b$ and $c$ .

[3]

II.

Write $h(x)$ in the form $2x-1+\dfrac{R}{x-3}$ .

[2]

Use the form found in part (a).

I.

Show that the graph of $y=h(x)$ does not intersect the line $y=x$ .

[2]

II.

Find the set of values of $x$ for which the vertical distance between the graph of $y=h(x)$ and its oblique asymptote is less than $0.1$ .

[3]

Question 48

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

In a model, a signed magnification $M$ depends on a position $x$ according to

M(x)=\frac{ax+b}{x^2-5x+4},\quad x\neq 1,4

It is known that $M(0)=\frac12$ and $M(2)=-3$ .

I.

Find the values of $a$ and $b$ .

[2]

II.

Find the $x$ -intercept of the graph of $y=M(x)$ .

[1]

III.

Express $M(x)$ in partial fractions.

[2]

Use the model from part (a).

I.

Find the exact value of $\displaystyle \int_5^8 M(x)\,\text{d}x$ .

[3]

II.

Find the value of $x>4$ for which $M(x)=1$ .

[2]

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Question 49

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let

f(x)=\frac{x^2+px+q}{x+1}, \quad x\neq -1

where $p$ and $q$ are constants. The graph of $y=f(x)$ has oblique asymptote $y=x+2$ and passes through the point $(1,4)$ .

I.

Find the value of $p$ .

[2]

II.

Find the value of $q$ .

[3]

Show that $f(x)=x+2+\dfrac{2}{x+1}$ .

[2]

Show that the graph of $y=f(x)$ has no $x$ -intercepts.

[2]

Find the coordinates of the points where the graph of $y=f(x)$ intersects the line $y=2x+1$ .

[4]

Question 50

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let

f(x)=\frac{4x}{x^2+4}

I.

State the domain of $f$ and the equation of the horizontal asymptote.

[2]

II.

Find the intercept with the coordinate axes.

[1]

Show that

f'(x)=\frac{4(4-x^2)}{(x^2+4)^2}

[3]

Find the coordinates and nature of the stationary points of the graph of $y=f(x)$ .

[4]

Solve $f(x)=\dfrac12$ .

[2]

Hence state the range of $f$ .

[2]

Question 51

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

For $k\in\mathbb{R}$ , define

f_k(x)=\frac{x+1}{x^2-4x+k}

I.

Write down the equation of the horizontal asymptote of the graph of $y=f_k(x)$ .

[1]

II.

Find the $y$ -intercept in terms of $k$ , where it exists.

[2]

Determine the values of $k$ for which the graph of $y=f_k(x)$ has no vertical asymptotes.

[3]

For $k=3$ , express $f_3(x)$ in partial fractions.

[3]

For $k=3$ , find the equations of the vertical asymptotes and the $x$ -intercept of the graph of $y=f_3(x)$ .

[3]

For $k=3$ , solve $f_3(x)>0$ .

[1]

Question 52

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Let

f(x)=\frac{2x-3}{x^2-4x+3}, \quad x\ne 1,3

I.

Find the equations of all asymptotes of the graph of $y=f(x)$ .

[3]

II.

Find the $x$ -intercept of the graph.

[1]

Express $f(x)$ in partial fractions.

[3]

Hence find

\int_4^8 f(x)\,dx

[3]

Using a GDC, find the coordinates of the points where the graph of $y=f(x)$ intersects the line $y=0.4$ .

[3]

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Question 53

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

For $a\in\mathbb{R}$ , define

f_a(x)=\frac{x+a}{x^2-2ax+9}

I.

Find the discriminant of the denominator, in terms of $a$ .

[1]

II.

Determine the set of values of $a$ for which the graph of $y=f_a(x)$ has no vertical asymptotes.

[2]

For the remainder of the question, let $a=5$ . Write down the equations of the vertical and horizontal asymptotes of the graph of $y=f_5(x)$ , and find its $x$ -intercept.

[3]

Find the coordinates of the stationary points of the graph of $y=f_5(x)$ .

[4]

Classify each stationary point as a local maximum or a local minimum.

[2]

Question 54

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

For $k\in\mathbb{R}$ , define

f_k(x)=\frac{x^2+kx+4}{x+2}, \quad x\ne -2

I.

Show that

f_k(x)=x+k-2+\frac{8-2k}{x+2}

[2]

II.

Write down the equation of the oblique asymptote of the graph of $y=f_k(x)$ .

[1]

Let $k=6$ . Find the equations of the asymptotes of the graph of $y=f_6(x)$ and the coordinates of its intercepts with the axes.

[4]

For $k=6$ , find the coordinates of the points where the graph of $y=f_6(x)$ intersects the line $y=3$ .

[2]

Determine the value of $k$ for which the graph of $y=f_k(x)$ has no vertical asymptote. Describe what happens at $x=-2$ for this value of $k$ .

[3]

Question 55

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $a\in\mathbb{R}$ , define

f_a(x)=\frac{ax+1}{x-a},\quad x\neq a

This question investigates a family of linear rational functions related to the reciprocal function.

I.

Write down the equations of the vertical and horizontal asymptotes of the graph of $y=f_a(x)$ .

[2]

II.

Show that $f_a$ is self-inverse on its domain.

[2]

Let $a=2$ .

I.

Find the fixed points of $f_2$ , where $f_2(x)=x$ .

[3]

II.

Find the intercepts of the graph of $y=f_2(x)$ with the coordinate axes.

[2]

III.

Sketch the graph of $y=f_2(x)$ , showing the asymptotes and fixed points.

[1]

Question 56

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A variable $x$ changes with time $t$ according to

\frac{\text{d}x}{\text{d}t}=\frac{(x-1)(x+3)}{x+1},\quad x(0)=2

This question uses partial fractions to solve the differential equation.

I.

Show that

\frac{x+1}{(x-1)(x+3)}\equiv \frac12\left(\frac{1}{x-1}+\frac{1}{x+3}\right)

[2]

II.

Hence show that

t=\frac12\ln\left(\frac{(x-1)(x+3)}{5}\right)

[2]

If you did not obtain the result in part (a)(ii), use it for this part.

I.

Express $x$ as a function of $t$ .

[3]

II.

Find the time when $x=10$ .

[2]

III.

Find $\displaystyle \lim_{t\to\infty}x(t)$ and interpret your result.

[1]

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Question 57

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $a\in\mathbb{R}$ , define

p_a(x)=\frac{x-a}{x^2+ax+a^2+1}

I.

Show that the graph of $y=p_a(x)$ has no vertical asymptotes for any real value of $a$ .

[2]

II.

Find the $x$ -intercept and the horizontal asymptote of the graph of $y=p_a(x)$ .

[2]

Let $a=1$ .

I.

Find the stationary points of $y=p_1(x)$ .

[3]

II.

Hence state the range of $p_1$ .

[2]

III.

Sketch $y=p_1(x)$ , showing the intercepts, horizontal asymptote and stationary points.

[1]

Question 58

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Consider

q(x)=\frac{x^2+px+q}{x+1},\quad x\neq -1

The graph of $y=q(x)$ has oblique asymptote $y=x+2$ and has a stationary point on the $x$ -axis.

I.

Find the values of $p$ and $q$ .

[3]

II.

Write the resulting function in the form $x+2+\dfrac{R}{x+1}$ .

[2]

Use the function found in part (a).

I.

Find the coordinates of both stationary points.

[3]

II.

Find the vertical distance between the graph and its oblique asymptote when $x=4$ .

[1]

III.

Sketch the graph, showing both asymptotes and both stationary points.

[1]

Question 59

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For constants $a$ , $b$ and $c$ , let

F(x)=\frac{x^2+bx+c}{x-a},\quad x\neq a

This question investigates the relationship between the remainder after division and the stationary points of $F$ .

AI.

Show that

F(x)=x+a+b+\frac{a^2+ab+c}{x-a}

[2]

AII.

Let $R=a^2+ab+c$ . Show that $F$ has two stationary points if and only if $R>0$ .

[2]

Assume that $R>0$ .

BI.

Find the coordinates of the stationary points in terms of $a$ , $b$ and $R$ .

[3]

BII.

Deduce that the midpoint of the two stationary points lies on the vertical asymptote.

[2]

BIII.

For $a=2$ , $b=-3$ and $c=8$ , find the stationary points of $F$ .

[1]

Question 60

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $A>0$ , consider the family of rational functions

H_A(x)=x+1+\frac{A}{x-2},\quad x\neq 2

The graphs all have the same vertical and oblique asymptotes.

I.

Write $H_A(x)$ as a single rational expression of the form $\dfrac{x^2+bx+c}{x-2}$ .

[2]

II.

Show that the stationary points of $H_A$ occur at $x=2\pm\sqrt A$ .

[2]

If you did not obtain the result in part (a)(ii), use it for this part.

I.

For $A=9$ , find the coordinates of the stationary points.

[2]

II.

For $A=9$ , solve $H_A(x)=10$ .

[2]

III.

Determine the value of $A$ for which the minimum value of $H_A(x)$ on the interval $x>2$ is $8$ .

[2]

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