Rational functions: Quadratic over linear

Error

There was an error loading the page.

Contributors

Ben Brawn

Feedback

From users who are members of Graphing and Polynomials :

Ben Brawn

said

Ready to use

7 years, 6 months ago

History

○

Ben Brawn 7 years, 6 months ago

Published this.

○

Ben Brawn 7 years, 6 months ago

Published this.

○

Ben Brawn 7 years, 6 months ago

Gave some feedback: Ready to use

○

Ben Brawn 7 years, 6 months ago

Created this as a copy of Right hyperbolas 2.

Name	Status	Author	Last Modified
Right hyperbolas	draft	Ben Brawn	03/03/2025 00:32
Jon's copy of Right hyperbolas	draft	Jon Stammers	17/11/2016 11:46
Asymptotes and intercepts	draft	Denis Flynn	28/02/2017 12:27
Right hyperbolas 2	draft	Ben Brawn	03/03/2025 00:33
Rational functions: Quadratic over linear	Ready to use	Ben Brawn	03/03/2025 00:33
riccardo's copy of Rational functions: Quadratic over linear	draft	riccardo belle'	10/09/2017 19:13
Maria's copy of Rational functions: Quadratic over linear	draft	Maria Aneiros	11/06/2019 05:18
Rational functions: Quadratic over linear	draft	Xiaodan Leng	11/07/2019 01:33

There are 5 other versions that do you not have access to.

Question statement

You are given the equation

$\simplify[all]{y=({b}x^2-{a*b+c}x+{a*c+d})/(x-{a})}.$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	-12
b	integer	-3
c	integer	-7
d	integer	-5
des	number	1789
xint	set	Set of 2 items

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(-12..12 except 0)

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `integer`

-12

→ Used by:

des
xint

Statement
Advice
"Part a)" → "Unnamed gap" - Minimum accepted value
"Part a)" → "Unnamed gap" - Maximum accepted value

Parts

Part a) Gap-fill
1. Gap Number entry
  Add
Part b) Gap-fill
1. Gap Mathematical expression
  Add
Part c) Choose several from a list
Part d) Gap-fill
1. Gap Number entry
  Add
Part e) Gap-fill
1. Gap Mathematical expression
  Add

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

The graph of this equation has a vertical asymptote at $x=$ Unnamed gap.

Advice

The vertical asymptote corresponds to the value of $x$ that results in attempting to divide by $0$ . For the equation

$\simplify[all]{y=({b}x^2-{a*b+c}x+{a*c+d})/(x-{a})}$

This means the equation of the vertical asymptote is $x=\var{a}$ .

The oblique asymptote corresponds to the value of $y$ that results from $x$ approaching infinity. Let's look at two methods that determine this limit:

Method 1 - Breaking up the fraction

We can rewrite the numerator of $\simplify[all]{({b}x^2-{a*b+c}x+{a*c+d})/(x-{a})}$ so that it includes multiples of the denominator. We do this term by term. We first write $\simplify{{b}x^2}$ in terms of $\simplify{x-{a}}$ :

$\simplify{{b}x^2={b}x(x-{a})+{a*b}x}$

and so the next term is now $\simplify[!collectNumbers]{{a*b}x-{a*b+c}x=-{c}x}$ . We now write this in terms of $\simplify{x-{a}}$ :

$\simplify{-{c}x=-{c}(x-{a})-{a*c}}$

and so our next term is now $\simplify[!collectNumbers]{-{a*c}+{a*c+d}={d}}$ . Now we can rewrite the numerator and break up our fraction:

$\begin{align}\simplify[all]{({b}x^2-{a*b+c}x+{a*c+d})/(x-{a})}&=\simplify{({b}x(x-{a})-{c}(x-{a})+{d})/(x-{a})}\\&=\simplify{({b}x(x-{a}))/(x-{a})-({c}(x-{a}))/(x-{a})+{d}/(x-{a})}\\&=\simplify{({b}x-{c})+{d}/(x-{a})}\end{align}$

Now as $x$ gets very large $\simplify{{d}/(x-{a})}$ gets very close to $0$ and so $y$ approaches $\simplify{{b}x-{c}}$ .

Method 2 - Polynomial long division

There is a procedure called polynomial long division, if you are comfortable with long division of numbers then it isn't too different. You can see an example of the procedure here. The result of the division (in this case $\simplify{{b}x-{c}}$ ) will be the oblique asymptote. The remainder will be $\var{d}$ , which 'remains' to be divided by $\simplify{x-{a}}$ . Just like in the method above, you will be able to say

$y=\simplify[all]{({b}x^2-{a*b+c}x+{a*c+d})/(x-{a})}=\simplify{({b}x-{c})+{d}/(x-{a})}$
so as

$x$ gets very large

$\simplify{{d}/(x-{a})}$ gets very close to

$0$ and so

$y$ approaches

$\simplify{{b}x-{c}}$ .

This means the equation of the horizontal asymptote is $y=\simplify{{b}x-{c}}$ .

Notice that from Method 1 and Method 2 above we found that our equation can be written as $\simplify{{b}x-{c}+{d}/(x-{a})}$ , and so we can see that $\var{d}$ is multiplying the fraction $\simplify{1/(x-{a})}$ . It is a general fact that because $\var{d}$ is positive the graph will be in the top right and bottom left parts of the plane. negative the graph will be in the top left and bottom right parts of the plane. We can see this as follows:

By substituting into the equation an $x$ value that is to the right of the vertical asymptote $x=\var{a}$ , we will find the resulting $y$ value is the oblique asymptote $y=\simplify{{b}x-{c}+{d}}$ .
By substituting into the equation an $x$ value that is to the left of the vertical asymptote $x=\var{a}$ , we will find the resulting $y$ value is the oblique asymptote $y=\simplify{{b}x-{c}+{d}}$ .

To find the $y$ -intercept, let $x=0$ and solve for $y$ :

$y$	$=$	$\displaystyle\simplify[!collectNumbers]{({b}0^2-{ab+c}0+{ac+d})/(0-{a})}$
	$=$	$\displaystyle\simplify[fractionNumbers]{{-c-d/a}}$

To find the $x$ -intercept, let $y=0$ and solve for $x$ (here we use the quadratic equation):

$0$	$=$	$\displaystyle\simplify{({b}x^2-{ab+c}x+{ac+d})/(x-{a})}$
$0$	$=$	$\displaystyle\simplify{({b}x^2-{ab+c}x+{ac+d})}$
$x$	$=$	$\displaystyle\frac{-(\var{-ab-c})\pm\sqrt{(\var{-ab-c})^2-4(\var{b})(\var{a*c+d})}}{2(\var{b})}$

Since this includes the square root of a negative number there are no $x$ -intercepts.

Therefore, the $x$ -intercept is $x=\simplify[fractionNumbers,simplifyFractions,unitDenominator]{({a*b+c})/{2*b}}$ .

Therefore, the $x$ -intercepts are $x=\simplify{({a*b+c}-sqrt({des}))/{2*b}}$ and $x=\simplify{({a*b+c}+sqrt({des}))/{2*b}}$ .

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Choose several from a list
Choose several from a list		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet

This question is used in the following exams:

Rational functions by Ben Brawn in Graphing and Polynomials.
Graphing and Polynomials by Chris Templet in Chris's workspace.

Loading...

Error

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ben Brawn 7 years, 6 months ago

Ben Brawn 7 years, 6 months ago

Ben Brawn 7 years, 6 months ago

Ben Brawn 7 years, 6 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

Loading...

Error

Rational functions: Quadratic over linear

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ben Brawn 7 years, 6 months ago

Ben Brawn 7 years, 6 months ago

Ben Brawn 7 years, 6 months ago

Ben Brawn 7 years, 6 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`