In Grade 10, you studied graphs of many different forms. In this chapter, you will learn a little more about the graphs of functions.
Functions of the form
$y=\frac{a}{x+p}+q$
This form of the hyperbolic function is slightly more complex than the form studied in Grade 10.
Investigation : functions of the form
$y=\frac{a}{x+p}+q$
On the same set of axes, plot the following graphs:
$a\left(x\right)=\frac{-2}{x+1}+1$
$b\left(x\right)=\frac{-1}{x+1}+1$
$c\left(x\right)=\frac{0}{x+1}+1$
$d\left(x\right)=\frac{1}{x+1}+1$
$e\left(x\right)=\frac{2}{x+1}+1$
Use your results to deduce the effect of
$a$ .
On the same set of axes, plot the following graphs:
$f\left(x\right)=\frac{1}{x-2}+1$
$g\left(x\right)=\frac{1}{x-1}+1$
$h\left(x\right)=\frac{1}{x+0}+1$
$j\left(x\right)=\frac{1}{x+1}+1$
$k\left(x\right)=\frac{1}{x+2}+1$
Use your results to deduce the effect of
$p$ .
Following the general method of the above activities, choose your own values of
$a$ and
$p$ to plot 5 different graphs of
$y=\frac{a}{x+p}+q$ to deduce the effect of
$q$ .
You should have found that the sign of
$a$ affects whether the graph is located in the first and third quadrants, or the second and fourth quadrants of Cartesian plane.
You should have also found that the value of
$p$ affects whether the
$x$ -intercept is negative (
$p>0$ ) or positive (
$p<0$ ).
You should have also found that the value of
$q$ affects whether the graph lies above the
$x$ -axis (
$q>0$ ) or below the
$x$ -axis (
$q<0$ ).
These different properties are summarised in
[link] . The axes of symmetry for each graph is shown as a dashed line.
Table summarising general shapes and positions of functions of the form
$y=\frac{a}{x+p}+q$ . The axes of symmetry are shown as dashed lines.
$p<0$
$p>0$
$a>0$
$a<0$
$a>0$
$a<0$
$q>0$
$q<0$
Domain and range
For
$y=\frac{a}{x+p}+q$ , the function is undefined for
$x=-p$ . The domain is therefore
$\{x:x\in \mathbb{R},x\ne -p\}$ .
We see that
$y=\frac{a}{x+p}+q$ can be re-written as:
This shows that the function is undefined at
$y=q$ . Therefore the range of
$f\left(x\right)=\frac{a}{x+p}+q$ is
$\left\{f\right(x):f(x)\in R,f(x)\ne q$ .
For example, the domain of
$g\left(x\right)=\frac{2}{x+1}+2$ is
$\{x:x\in \mathbb{R},x\ne -1\}$ because
$g\left(x\right)$ is undefined at
$x=-1$ .
We see that
$g\left(x\right)$ is undefined at
$y=2$ . Therefore the range is
$\left\{g\right(x):g(x)\in (-\infty ,2)\cup (2,\infty \left)\right\}$ .
Domain and range
Determine the range of
$y=\frac{1}{x}+1$ .
Given:
$f\left(x\right)=\frac{8}{x-8}+4$ . Write down the domain of
$f$ .
Determine the domain of
$y=-\frac{8}{x+1}+3$
Intercepts
For functions of the form,
$y=\frac{a}{x+p}+q$ , the intercepts with the
$x$ and
$y$ axis are calculated by setting
$x=0$ for the
$y$ -intercept and by setting
$y=0$ for the
$x$ -intercept.
Given:
$h\left(x\right)=\frac{1}{x+4}-2$ . Determine the coordinates of the intercepts of
$h$ with the x- and y-axes.
Determine the x-intercept of the graph of
$y=\frac{5}{x}+2$ . Give the reason why there is no y-intercept for this function.
Asymptotes
There are two asymptotes for functions of the form
$y=\frac{a}{x+p}+q$ . They are determined by examining the domain and range.
We saw that the function was undefined at
$x=-p$ and for
$y=q$ . Therefore the asymptotes are
$x=-p$ and
$y=q$ .
For example, the domain of
$g\left(x\right)=\frac{2}{x+1}+2$ is
$\{x:x\in \mathbb{R},x\ne -1\}$ because
$g\left(x\right)$ is undefined at
$x=-1$ . We also see that
$g\left(x\right)$ is undefined at
$y=2$ . Therefore the range is
$\left\{g\right(x):g(x)\in (-\infty ,2)\cup (2,\infty \left)\right\}$ .
From this we deduce that the asymptotes are at
$x=-1$ and
$y=2$ .
Asymptotes
Given:
$h\left(x\right)=\frac{1}{x+4}-2$ .Determine the equations of the asymptotes of
$h$ .
Write down the equation of the vertical asymptote of the graph
$y=\frac{1}{x-1}$ .
Sketching graphs of the form
$f\left(x\right)=\frac{a}{x+p}+q$
In order to sketch graphs of functions of the form,
$f\left(x\right)=\frac{a}{x+p}+q$ , we need to calculate four characteristics:
domain and range
asymptotes
$y$ -intercept
$x$ -intercept
For example, sketch the graph of
$g\left(x\right)=\frac{2}{x+1}+2$ . Mark the intercepts and asymptotes.
We have determined the domain to be
$\{x:x\in \mathbb{R},x\ne -1\}$ and the range to be
$\left\{g\right(x):g(x)\in (-\infty ,2)\cup (2,\infty \left)\right\}$ . Therefore the asymptotes are at
$x=-1$ and
$y=2$ .
The
$y$ -intercept is
${y}_{int}=4$ and the
$x$ -intercept is
${x}_{int}=-2$ .
Graphs
Draw the graph of
$y=\frac{1}{x}+2$ . Indicate the horizontal asymptote.
Given:
$h\left(x\right)=\frac{1}{x+4}-2$ . Sketch the graph of
$h$ showing clearly the asymptotes and ALL intercepts with the axes.
Draw the graph of
$y=\frac{1}{x}$ and
$y=-\frac{8}{x+1}+3$ on the same system of axes.
Draw the graph of
$y=\frac{5}{x-2,5}+2$ . Explain your method.
Draw the graph of the function defined by
$y=\frac{8}{x-8}+4$ . Indicate the asymptotes and intercepts with the axes.
End of chapter exercises
Plot the graph of the hyperbola defined by
$y=\frac{2}{x}$ for
$-4\le x\le 4$ . Suppose the hyperbola is shifted 3 units to the right and 1 unit down. What is the new equation then ?
Based on the graph of
$y=\frac{1}{x}$ , determine the equation of the graph with asymptotes
$y=2$ and
$x=1$ and passing through the point (2; 3).
Questions & Answers
find the 15th term of the geometric sequince whose first is 18 and last term of 387
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Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change .
maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
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Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
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