Graphing Rational Functions
Graphing Asymptotic Functions
Graph Requirements:
You are required to graph six points for each function.
Three points on the left side of the vertical asymptote (VA).
Three points on the right side of the vertical asymptote.
At most, you will plot nine points total, which includes:
Left - Closest to VA
Middle - Closer to the origin, where applicable
Right - Closest to VA
The fundamental requirement is to ensure points are as close to the vertical asymptote as possible while still capturing the overall curve.
Asymptotes
Vertical Asymptote (VA): This is a vertical line that a graph approaches but never touches or crosses.
Horizontal Asymptote (HA): This is a horizontal line that the graph approaches as the x-values go to positive or negative infinity.
Plotting Asymptotes:
The VA and HA must be clearly marked on the graph.
Use dotted lines for the asymptotes, extending them sufficiently so they do not appear cut off.
For vertical asymptotes, note the specific x-coordinate.
Example Asymptote Locations:
Vertical Asymptote (VA) for a specific function at x = -1.
Horizontal Asymptote (HA) at y = 2.
Points to Plot
Required Points:
Always start by identifying the first six points to plot, on either side of the VA:
Example: If the VA is at x = -1, you should choose x-coordinates such as -2, -3, 0, 1, 2, and 3.
This maintains the ratio of points on both sides of the VA.
Additionally, you can add more points if needed, ensuring a smooth and correct representation of the curve.
Graphing Steps
Identify Asymptotes:
For instance, if a graph has a vertical asymptote at x = -1 and a horizontal asymptote at y = 2, you will:
Draw a vertical dashed line at x = -1.
Draw a horizontal dashed line at y = 2.
Determine & Plot Points:
Calculate y-values for chosen x-values close to the vertical asymptote.
Example Calculations:
For x = -2: Find y (depends on specific function).
For x = 0: Find y (depends on specific function).
Mark points on the graph with coordinates, such as (-2, y1), (0, y0), etc.
Connecting Points:
Draw the curve smoothly connecting all plotted points while approaching the asymptotes.
Ensure you lift the pencil at holes in the function where applicable.
Holes in Functions
A hole occurs in a rational function when both the numerator and denominator share a common factor that can be canceled out, resulting in a point where the function is undefined.
Indicate holes with an open circle on the graph at their corresponding coordinates.
Example: A hole at (1/3, -17/29) indicates the function is undefined at that point.
Domain and Range
Domain: The set of all possible x-values for which the function is defined, excluding points that yield division by zero or holes.
Example Determination: “Where does the domain get stopped by some imaginary line?” Typically halted at the vertical asymptote.
Range: The set of possible y-values, which may be limited due to horizontal asymptotes.
Not all functions have a defined range due to presence of asymptotes.
End Behavior
When considering end behavior as x approaches infinity or negative infinity, observe:
Functions with horizontal asymptotes approach a specific y-value.
The function behavior may differ when there is a presence of vertical asymptotes versus horizontal ones.
Key Concept Differentiations:
Functions with horizontal asymptotes (HA) often converge to a numeric value as x approaches infinity, while those with vertical asymptotes (VA) will tend toward infinity or negative infinity, indicating the function is unbounded in that direction.
Important Notes for Testing
You will need to perform these graphing tasks without graphing software or material, primarily by hand.
Ensure clarity and precision in your plots, understanding the requirements to avoid losing points.
Testing Schedule:
Upcoming test on higher order functions focusing on graphing and understanding asymptotes.
Review quiz number 8 will cover the topic of expanding and simplifying rational expressions.