Graphing Rational Functions: Holes In Rational Functions – Mac Grapher, A Graphing Calculator
Graphing Rational Functions: Holes In Rational Functions – Mac Grapher, A Graphing Calculator

Here are a few recommended readings before getting started with this lesson.

A rational function is a function that contains a rational expression. Any function that can be written as the quotient of two polynomial functions and is a rational function.

The graph of a rational function can be a smooth continuous curve, or it can have jumps, breaks, or holes. By looking at the graph of a function, functions can be categorized into two groups: continuous or discontinuous.

A point of discontinuity of a function is a point with the coordinate that makes the function value undefined. A point of discontinuity can also be considered as an excluded value of the function rule.

holein the graph.

When a function cannot be redefined so that the point of discontinuity becomes a valid input, it is called a non-removable discontinuity. Consider, for example, Since is not in the domain of the function, there is a point of discontinuity at

Total Annual Income Population

Substitute expressions

is undefined

On the way to the pencil factory, LaShay noticed that road maintenance work is being carried out to fill a

hole

on the road.

Notice that it is also a removable discontinuity because is the common factor in the numerator and denominator.

This graph also has a removable discontinuity at because is the common factor in the numerator and denominator of the rational function.

An asymptote of a graph is an imaginary line that the graph gets close to as goes to plus or minus infinity or a particular number. For example, the graph of the rational function has two asymptotes — the axis and the axis.

Analyzing the diagram, the following can be observed.

To identify the vertical asymptotes, the function should be in its simplest form. That is, if and have common factors, the function should be simplified first. The vertical asymptotes will occur at the zeros of the denominator. Check out the following examples.

The degrees of polynomials in the numerator and denominator of a rational function will determine if the graph has a horizontal asymptote.

Horizontal Asymptote If there is no horizontal asymptote.
If the horizontal asymptote is
If the horizontal asymptote is

Vertical Asymptotes: and
Horizontal Asymptote:

Factor Theorem

The expression is a factor of a polynomial if and only if the value is a zero of the related polynomial function.

Cross out common factors

Cancel out common factors

Asymptote
none

An asymptote can be neither vertical nor horizontal. It can be a slanted line. In such cases, it is called an oblique asymptote or slant asymptote.

Multiply by

Subtract down

The students spent about hours at the factory. At the end of this period, LaShay’s teacher made a survey about the trip and drew a graph relating the students’ interest in the trip and the time elapsed at the factory. The greater the value, the greater the interest of the students.

Vertical Asymptotes:
Oblique Asymptote:

Subtract fractions

Commutative Property of Addition

Multiply by

Subtract down

Multiply by

Subtract down

holeat Also, the graph has a vertical asymptote at because is not included in the domain.

Asymptote Asymptote Type
Horizontal
None None
Horizontal
the quotient of the polynomials with no remainder Oblique

Multiply by

Subtract down

Multiply by

Subtract down

The given function has one vertical asymptote. Therefore, its graph will consist of two parts, one to the left and the other to the right of the vertical asymptote. These parts may lie above or below the oblique asymptote.

Some additional points are needed to get a good sense of the shape of the graph. Make sure to only use values included in the domain of the function.

It is almost done. Plot the points and imagine how the shape of the graph should look!

As can be seen, one part of the graph will lie to the left of the vertical asymptote and below the oblique asymptote. The other part of the graph will be to the right of the vertical asymptote and above the oblique asymptote.

The graph can now be drawn by connecting the points with a smooth curve. It must approach the asymptotes. Do not forget to plot the hole at

The factory sells pencils in a rectangular box. The dimensions of the box are shown in the diagram.

Add and subtract terms

Substitute expressions

holesin the graph. Recall that if the real number is not included in the domain, there is a vertical asymptote at In this case, there are two vertical asymptotes, one at and the other at

Asymptote
none
the quotient of the polynomials with no remainder

Calculate power

Zero Property of Multiplication

Identity Property of Addition

Now, make a table of values to graph the given function.

Finally, the graph of the function can be drawn by plotting the found points and connecting them with a smooth curve. Recall that a rational function can cross the horizontal asymptote but cannot cross the vertical asymptotes. In this case, the horizontal asymptote is crossed.

Recall the Factor Theorem.

has a factor if and only if

Substitute values

Identity Property of Multiplication

Calculate power

Split into factors

Calculate root

Simplify quotient

Interpretation: In about years, there will be no water left in the pond.

Equation:

Now, push to draw the function.

It appears that only one part of the graph is shown, but it is not possible to know that until the size of the viewing window is changed.

The graph does not seem to have any zeros. However, note that as increases in the first quadrant, the graph gets closer to the axis and could cross it. To check this, zoom in on this part by changing the window settings. Push and change the settings as shown below. Then push once more to draw the equation with these new settings.

Now it can be seen that the graph intersects the axis and there is a zero. To find it, use the zero option in the calculator. It can be found by pressing and then

After selecting the

zero

option, choose left and right boundaries for the zero. Finally, the calculator asks for a guess where the zero might be. After that, it will calculate the exact point.

The function intersects the axis at about Since represents the numbers of years that have passed since pumping starts, the intercept means that in about years, there will be no water left in the pond.

Asymptote Asymptote Type
Horizontal
None None
Horizontal
the quotient of the polynomials with no remainder Oblique

Multiply by

Subtract down

Multiply by

Subtract down

Asymptote Asymptote Type
Horizontal
None None
Horizontal
the quotient of the polynomials with no remainder Oblique

Multiply by

Subtract down

Multiply by

Subtract down

A table of values will be used to get a rough idea of the shape of the graph.

Plot the points and imagine how the shape of the graph should look!

Finally, draw the graph by connecting the points. It must approach, but not cross, the asymptotes.

As shown, the range does not contain all real numbers. It appears that the minimum point of the part of the graph in the first quadrant is Notice also that the graph is symmetric about the point of intersection of the asymptotes, Using this fact, the maximum point of the other part can be identified.

A graphing calculator can be used to check the answers. The graph of the function will be drawn first. Press the button and type the equation in the first row.

By pushing the calculator will draw the equation. For this function to be visible on the screen, re-size the standard window by pushing the button. Change the settings to a more appropriate size and then push

Next, the local maximum and local minimum will be found. To do so, push then and choose the

maximum

option.

When using the

maximum

feature, choose the left and right bounds. The calculator will then provide a best guess as to where the maximum might be.

The maximum point of the graph in the third quadrant is To find the minimum, repeat the same process but choose the

minimum

option.

The minimum point of the part in the first quadrant is As a result, the range is and the domain is all real numbers except

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