Right hyperbolas 2

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Ben Brawn

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Ben Brawn 7 years, 6 months ago

Created this as a copy of Right hyperbolas.

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

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Question statement

You are given the equation

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

Give any introductory information the student needs.

			Name	Type	Generated Value

k	integer	-6
a	integer	-12
b	integer	-1

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(-12..12 except 0)

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Description

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

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Generated value: `integer`

-6

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
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Part b) Gap-fill
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Part c) Choose several from a list
Part d) Gap-fill
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Part e) Gap-fill
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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

$\simplify[all]{y=({b}x+{k-a*b})/(x-{a})}$

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

The horizontal 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 $\simplify[all]{({b}x+{k-a*b})/(x-{a})}$ so that it includes a multiple of the denominator as such:

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

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

Method 2 - Dividing by the dominant term

Since $\simplify{({b}x+{k-a*b})/(x-{a})}$ is a fraction be can rewrite it by dividing the numerator and denominator by the same value, in this case we will divide by $x$ :

$\begin{align}\simplify[all]{({b}x+{k-a*b})/(x-{a})}&=\simplify[all]{({b}x/x+{k-a*b}/x)/(x/x-{a}/x)}\\&=\simplify[all]{({b}+{k-a*b}/x)/(1-{a}/x)}\end{align}$

Now as $x$ gets very large both $\simplify{{k-a*b}/x}$ and $\simplify{{a}/x}$ gets very close to zero and so $y$ approaches $\var{b}$ .

This means the equation of the horizontal asymptote is $y=\var{b}$ .

Notice that from Method 1 above we found that our equation can be written as $\simplify[all]{y={b}+{k}/(x-{a})}$ , and so we can see that $\var{k}$ is multiplying the fraction $\simplify[all]{1/(x-{a})}$ . It is a general fact that because $\var{k}$ 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 horizontal asymptote $y=\var{b}$ .
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 horizontal asymptote $y=\var{b}$ .

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

$y$	$=$	$\simplify[!collectNumbers]{({b}0+{k-a*b})/(0-{a})}$
	$=$	$\simplify[fractionNumbers]{{-k/a+b}}$

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

$0$	$=$	$\simplify{({b}x+{k-a*b})/(x-{a})}$
$0$	$=$	$\simplify{({b}x+{k-a*b})}$
$\simplify{{-b}x}$	$=$	$\var{k-a*b}$
$x$	$=$	$\simplify[fractionNumbers]{{-k/b+a}}$

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This question is used in the following exams:

Graphing: Right hyperbolas by Ben Brawn in Graphing and Polynomials.
Graphing and Polynomials by Chris Templet in Chris's workspace.

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Ben Brawn 7 years, 6 months ago

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Generated value: `integer`

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Right hyperbolas 2

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Ben Brawn 7 years, 6 months ago

Error in variable testing condition

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Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`