Identifying Key Features
Graphing any function, especially a rational one, requires a keen eye for its key features. These features not only dictate the overall shape and behaviour of the graph but also provide crucial points of reference:
- Intercepts: Intercepts are foundational to sketching the graph of a function. They are the points where the graph intersects the axes.
- X-intercepts: These are the points where the graph crosses the x-axis. They can be found by setting the numerator of the rational function to zero and solving for x.
- Y-intercepts: These are the points where the graph crosses the y-axis. They can be determined by setting x = 0 in the function and solving for y.
- Asymptotes: Asymptotes are lines that the graph of the function approaches but never actually touches or crosses.
- Vertical Asymptotes: These occur where the denominator of the rational function is zero, but the numerator isn't. The function's value will tend towards infinity (or negative infinity) as it approaches a vertical asymptote.
- Horizontal Asymptotes: These indicate the value that the function approaches as x tends towards positive or negative infinity. The position and existence of a horizontal asymptote are determined by comparing the degrees of the polynomials in the numerator and denominator.
- Holes or Removable Discontinuities: These are points on the graph where the function doesn't have a value. They occur when both the numerator and the denominator are zero. Unlike vertical asymptotes, the function doesn't tend towards infinity at these points.
Behaviour Near Asymptotes
The behaviour of a rational function near its asymptotes is a defining feature of its graph:
- Vertical Asymptotes: As the function approaches a vertical asymptote, its value will either shoot up towards positive infinity or plummet down towards negative infinity. The exact behaviour can be determined by analysing the sign of the function on either side of the asymptote.
- Horizontal Asymptotes: The function will get closer and closer to the horizontal asymptote as x moves towards positive or negative infinity. It's worth noting that unlike vertical asymptotes, the graph of a rational function can, and often does, cross a horizontal asymptote.
Sketching Graphs
With a solid understanding of the key features and behaviours of a rational function, you can sketch its graph with confidence:
1. Plot Intercepts: Begin by marking the x and y-intercepts on your graph. These points provide a foundational framework for your sketch.
2. Draw Asymptotes: Use dashed lines to represent the vertical and horizontal asymptotes. These lines serve as boundaries that the graph will approach but never cross (with the exception of horizontal asymptotes, which the graph can cross).
3. Evaluate Near Asymptotes: Choose a few x-values close to each vertical asymptote and calculate the corresponding y-values. This will give you a clearer picture of how the function behaves near its asymptotes.
4. Sketch the Curve: With the above points and lines as a guide, you can now sketch the curve of the function. Ensure that your sketch accurately reflects the function's behaviour, especially near its asymptotes.
5. Identify Holes: If the function has any removable discontinuities, mark them on your graph. These points are where the function is undefined.
Example Question: Consider the rational function R(x) = (x2 - 4) / (x2 - 2x). Sketch its graph.
Solution:
- The x-intercepts are found by setting the numerator to zero: x2 - 4 = 0, which gives x = 2 and x = -2.
- The y-intercept is found by setting x = 0: R(0) = 4/0, which means there isn't a y-intercept.
- The denominator x2 - 2x can be factored as x(x - 2). This gives potential vertical asymptotes at x = 0 and x = 2.
- The graph will approach the x-axis as x approaches infinity, so y = 0 is the horizontal asymptote.
- Using the above features, you can sketch the graph of R(x).
FAQ
To determine the behaviour of a rational function near a vertical asymptote, one can analyse the sign of the function on either side of the asymptote. By plugging in values slightly less than and slightly greater than the x-coordinate of the vertical asymptote, one can determine whether the function is heading towards positive infinity or negative infinity. This method gives insight into how the function behaves as it approaches the asymptote from the left and the right.
A horizontal asymptote describes the behaviour of a function as the input (x-value) approaches positive or negative infinity. It gives the y-value that the function is getting closer to, but it doesn't prohibit the function from reaching or crossing that value at finite x-values. In contrast, a vertical asymptote represents an x-value where the function is undefined and approaches positive or negative infinity. The function can never have a value at this x-coordinate, so the graph can never cross a vertical asymptote.
Holes, or removable discontinuities, in the graph of a rational function occur when both the numerator and the denominator have a common factor that can be cancelled out. At the x-value where this factor equals zero, the function is undefined, resulting in a hole. It's termed 'removable' because if we were to factorise and simplify the function, the common factor would cancel out, and the hole would be "removed", leaving a continuous function at that point.
A horizontal asymptote is a horizontal line that the graph of a function approaches as the input (x-value) heads towards positive or negative infinity. It's typically present when the degrees of the polynomials in the numerator and denominator of a rational function are equal. On the other hand, a slant (or oblique) asymptote is a straight line that isn't horizontal, which the graph of the function approaches. It occurs when the degree of the polynomial in the numerator is exactly one more than the degree of the polynomial in the denominator. The slant asymptote can be found using polynomial long division.
Yes, there are several tools and techniques to aid in sketching the graph of a rational function. Apart from the manual methods of finding intercepts, asymptotes, and analysing behaviour near asymptotes, technology can be a great aid. Graphing calculators and software like Wolfram Alpha, Desmos, and GeoGebra allow for accurate plotting of rational functions. These tools provide visual representations, making it easier to understand the function's behaviour across its domain. However, it's essential to understand the underlying concepts and not solely rely on technology.
Practice Questions
To find the x-intercepts, we set the numerator 2x3 - x2 + 4 to zero and solve for x. The y-intercept is found by setting x = 0 in the function. The vertical asymptotes are determined by setting the denominator x2 - 5x + 6 to zero. The horizontal asymptote can be determined by observing the degrees of the polynomials in the numerator and denominator. Since the degree of the numerator is greater than the degree of the denominator by one, there is no horizontal asymptote. Instead, there is a slant (or oblique) asymptote, which can be found by performing polynomial long division.
The graph of the function can be sketched using the previously identified features. Near the vertical asymptotes, the function will either head towards positive infinity or negative infinity, depending on the coefficients and structure of the function. The behaviour near the slant asymptote is such that as x moves towards positive or negative infinity, the graph of the function will get closer and closer to this slant asymptote. The exact behaviour can be determined by analysing the function's values on either side of the asymptotes and observing the graph. The graph provides a visual representation of the function's behaviour across its domain.
