Unit 2 - Functions
Required Prior Knowledge
Questions
Perform the following divisions to find the quotient and remainder, writing your answers in the form$$\frac{f\left(x\right)}{D\left(x\right)}=Q\left(x\right)+\frac{R}{D\left(x\right)}$$
a) \(\left(x^{2}+3x+5\right)\div\left(x+1\right)\)
b) \(\left(2x^{2}-5x+7\right)\div\left(x-2\right) \)
Solutions
Get Ready
Questions
Without using your calculator find the asymptotes and axes intercepts of the following functions, and hence, sketch the graph.
a) \(y=\frac{3x+2}{5-2x}\)
b) \(y=\frac{4-x}{x^{2}+3x-10}\)
Solutions
Investigation
Consider the function$$y=\frac{x^{2}+3x+5}{x+1}$$State the equation of the vertical asymptote.
From the Required Prior Knowledge we saw that $$\frac{x^{2}+3x+5}{x+1}=\left(x+2\right)+\frac{3}{x+1}$$Use the applet below to reveal the graph of \(y=\frac{x^{2}+3x+5}{x+1}\) and \(y=x+2\) on the same axes. What do you notice?
Explain why the graph of \(y=\frac{x^{2}+3x+5}{x+1}\) approaches the line \(y=x+2\) as \(x\to\infty\).
Go through the same steps again with the function$$y=\frac{2x^{2}-5x+7}{x-2}$$You can change the functions in the graph above to see the graphs.
Notes
Recall that an asymptote is a line that a function approaches as \(x\to\infty\).
If an asymptote is neither vertical nor horizontal it is called an oblique asymptote.
To find the oblique asymptote of $$y=\frac{ax^{2}+bx+c}{dx+e}$$ we perform a polynomial division to get the function in the form$$y=px+q+\frac{r}{dx+e}$$, where:
The quotient of the division is \(px+q\)
The remainder of the division is \(r\)
The function is undefined when \(dx+e=0\) so the vertical asymptote is \(x=-\frac{e}{d}\).
As \(x\to\infty\), we see that \(dx+e\to\infty\) meaning \(\frac{r}{dx+e}\to 0\) and so the graph gets closer to \(y=px+q\).
So the oblique asymptote is given by $$y=px+q$$which is the quotient when the polynomial division is performed.
Examples and Your Turns
Example
Consider the function$$f\left(x\right)=\frac{2x^{2}+3x-9}{x+2}$$
a) Find the vertical asymptote of \(y=f\left(x\right)\)
b) Find the axes intercepts
c) Find the oblique asymptote
d) Sketch the graph
Your Turn
Consider the function$$f\left(x\right)=\frac{x^{2}+3x-4}{x-2}$$
a) Find the vertical asymptote of \(y=f\left(x\right)\)
b) Find the axes intercepts
c) Find the oblique asymptote
d) Sketch the graph
Your Turn
Consider the function$$f\left(x\right)=\frac{x^{2}-3x-4}{x+2}$$
a) Find the vertical asymptote of \(y=f\left(x\right)\)
b) Find the axes intercepts
c) Find the oblique asymptote
d) Sketch the graph
Your Turn
Consider the function$$f\left(x\right)=\frac{2x^{2}+x-3}{x+2}$$
a) Find the vertical asymptote of \(y=f\left(x\right)\)
b) Find the axes intercepts
c) Find the oblique asymptote
d) Sketch the graph
Your Turn
A function has a vertical asymptote at \(x=4\) and an oblique asymptote at \(y=x+3\). The function passes through the origin.
Find a possible equation for the function.
Your Turn
The function \(f\left(x\right)=\frac{x^{2}+ax+b}{x+c}\) has an oblique asymptote of \(y=x-2\), a vertical asymptote of \(x=-1\) and a \(y\)-intercept at \(\left(0,3\right)\).
Find the values of \(a\), \(b\) and \(c\).
Challenge
Find a function with a parabola as an oblique asymptote.
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.
Taking it Deeper
Conceptual Questions to Consider
Why does the remainder of the division become insignificant as \(x\to\infty\)?
If the degree of the the numerator and the denominator are equal, what types of asymptotes will there be?
Can a rational function have both a horizontal asymptote and an oblique asymptote? Explain.
Common Mistakes / Misconceptions
The most common problem is a mistake in the polynomial division.
Sketching the graph is still difficult with all the information - focus on where it approaches the asymptotes.
Connecting This to Other Skills
This skill builds directly on Rational Functions (2.9).
You need to be able to divide Polynomials (2.19) to find oblique asymptotes.
The idea of asymptotes will appear again when we look at Limits and Continuity (4.2).
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?