Cubic and reciprocal functions

Identify x-intercepts and end behaviour of cubic functions in factored form, sketch and analyse reciprocal functions of the form

f(x) = k/x

, and match equations to graphs by justifying key features.

Worked examples

Finding x-intercepts and end behaviour of a cubic

Straightforward

Problem

For

f(x) = (x + 2)(x - 1)(x - 4)

, state the x-intercepts and describe the end behaviour.

Graphing a reciprocal function

Moderate

Problem

For

f(x) = \dfrac{-3}{x}

, state the asymptotes, domain and range, and which quadrants the graph occupies.

Finding a cubic equation from graph features

Challenging

Problem

A cubic graph has x-intercepts at

x = -2

x = 0

and

x = 3

, and as

x \to +\infty

y \to -\infty

. Find the equation of the cubic in fully factored form.

Practise

Open Math

Cubic and reciprocal functions

Algebra and Functions · MAV-11-01

Name:

Date:

Q1Straightforward

The function

f(x) = (x - 2)(x + 3)(x - 5)

has three x-intercepts. What is the largest x-intercept?

Q2Straightforward

For

f(x) = (x + 1)(x - 2)(x - 4)

, which statement correctly describes the end behaviour?

A.As $x \to +\infty$ , $f(x) \to +\infty$ and as $x \to -\infty$ , $f(x) \to -\infty$
B.As $x \to +\infty$ , $f(x) \to -\infty$ and as $x \to -\infty$ , $f(x) \to +\infty$
C.As $x \to \pm\infty$ , $f(x) \to +\infty$
D.As $x \to \pm\infty$ , $f(x) \to -\infty$

Q3Straightforward

The function

f(x) = (x + 1)(x - 3)(x + 5)

has three x-intercepts. What is their sum?

Q4Straightforward

x \to -\infty

, for which function does

f(x) \to +\infty

A. $f(x) = (x - 1)(x + 2)(x - 3)$
B. $f(x) = -(x - 1)(x + 2)(x - 3)$
C. $f(x) = x^2(x - 1)$
D. $f(x) = (x + 3)^3$

Q5Moderate

For

f(x) = \dfrac{3}{x}

, what are the equations of the asymptotes?

A. $x = 0$ and $y = 0$
B. $x = 3$ and $y = 3$
C. $x = 0$ and $y = 3$
D. $x = 3$ and $y = 0$

Q6Moderate

For

f(x) = \dfrac{-2}{x}

, in which quadrants does the graph lie?

A.Quadrants I and III
B.Quadrants II and IV
C.Quadrants I and II
D.Quadrants III and IV

Q7Moderate

What is the domain of

f(x) = \dfrac{5}{x}

A.All real numbers
B.All real numbers except $x = 0$
C. $x > 0$
D. $x \geq 5$

Q8Moderate

The graph of

f(x) = \dfrac{k}{x}

passes through the point

(2, -6)

. What is the value of

k

Q9Challenging

A cubic function has x-intercepts at

x = -3

x = 0

and

x = 4

, and its graph rises from bottom-left to top-right. Which equation matches?

A. $f(x) = x(x + 3)(x - 4)$
B. $f(x) = x(x - 3)(x + 4)$
C. $f(x) = -x(x + 3)(x - 4)$
D. $f(x) = x(x + 3)(x + 4)$

Q10Challenging

A reciprocal function has its graph in Quadrants II and IV and passes through

(-1, 5)

. Which equation fits?

A. $f(x) = \dfrac{5}{x}$
B. $f(x) = \dfrac{-5}{x}$
C. $f(x) = \dfrac{1}{5x}$
D. $f(x) = \dfrac{-1}{5x}$

Q11Challenging

The graph of

y = \dfrac{k}{x}

lies in Quadrants II and IV and passes through

(-3, 7)

. What is the value of

k

Q12Challenging

Which cubic equation has a negative leading coefficient and x-intercepts at

x = -1

x = 2

and

x = 5

A. $f(x) = (x + 1)(x - 2)(x - 5)$
B. $f(x) = -(x + 1)(x - 2)(x - 5)$
C. $f(x) = -(x - 1)(x + 2)(x + 5)$
D. $f(x) = (x - 1)(x + 2)(x + 5)$

Worked solutions and answers at openmath.au/year-11/advanced/working-with-functions/cubic-and-reciprocal-functions