Chapter 2.7, Problem 67E

To determine

To find the asymptotes and holes of the function $f(x)=\frac{2x^2-5x+2}{2x^2-x-6}$ and verifying them using graph.

Answer to Problem 67E

  $x=-1.5,y=1$

Explanation of Solution

Given:

Function: $f(x)=\frac{2x^2-5x+2}{2x^2-x-6}$

Calculation:

  $\begin{array}{l}\Rightarrow f(x)=\frac{2x^2-5x+2}{2x^2-x-6}\\ \Rightarrow f(x)=\frac{2x^2-4x-x+2}{2x^2-4x+3x-6}\\ \Rightarrow f(x)=\frac{2x(x-2)-1(x-2)}{2x(x-2)+3(x-2)}\\ \Rightarrow f(x)=\frac{(x-2)(2x-1)}{(x-2)(2x+3)}\\ \Rightarrow f(x)=\frac{2x-1}{2x+3}\end{array}$

Intercepts:

Let $f(x)=y.$

  $\Rightarrow y=\frac{2x-1}{2x+3}$

To find x intercepts, put y = 0,

  $\begin{array}{l}\Rightarrow y=0\\ \Rightarrow \frac{2x-1}{2x+3}=0\\ \Rightarrow 2x-1=0\\ \Rightarrow 2x=1\\ \Rightarrow x=\frac{1}{2}\end{array}$

So, x intercept is (0.5, 0).

To find y intercepts, put x = 0,

  $\begin{array}{l}\Rightarrow y=\frac{2(0)-1}{2(0)+3}\\ \Rightarrow y=\frac{-1}{3}\\ \Rightarrow y=-0.333\end{array}$

So, y intercept is (0, 0.333).

Asymptotes:

Vertical asymptotes:

To find vertical asymptotes, put denominator of the given function equal to zero.

  $\begin{array}{l}\Rightarrow 2x+3=0\\ \Rightarrow x=-\frac{3}{2}\\ \Rightarrow x=-1.5\end{array}$

Horizontal asymptotes:

As the degree of numerator is equal to the degree of the denominator, the horizontal asymptote is

  $\begin{array}{l}\Rightarrow y=\frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}\\ \Rightarrow y=\frac{2}{2}\\ \Rightarrow y=1\end{array}$

Slant asymptotes:

As the degree of numerator is equal to the degree of the denominator, the slant asymptote does not exist for given function.

Holes:

Here, the given function contains one common factor in numerator and denominator i.e.., $(x-2).$

To find hole of the function, equate common factor equal to zero.

  $\begin{array}{l}\Rightarrow x-2=0\\ \Rightarrow x=2\end{array}$

So, the hole of function is $x=-3.$

Calculation for graph:

Consider $f(x)=\frac{2x^2-5x+2}{2x^2-x-6}$

Values of x	Values of f (x)
0	-0.333
1	0.2
-1	-3
2	Undefined
-2	5

By taking different values of x, the graph can be plotted.

Graph:

  

Interpretation:

By observing graph, it is clear that

The x intercept is (0.5, 0),

The y intercept is (0, 0.333),

The vertical asymptotes are $x=-1.5$ ,

The horizontal asymptote is $y=1.$