Modulus Functions
The modulus sign, as mentioned in our section on algebra, means the numerical value of the function is taken. For example, |2| = 2, and |-2| = 2. In graphing, to take the modulus of a function is to:
1. The parts of the graph above the x-axis does not change.
2. The parts of the graph below to x-axis is reflected to above it.
Graph of y = |x|.
Sketch the graph of y = |x2 - 4x - 5|.
Solution:
Graph of y = x2 - 4x - 5.
Rational Functions
Rational functions are those of the form:
y = f(x)/g(x)
Apart from finding the x-intercepts, y-intercepts and turning points of rational functions (which we have described in our Techniques section), an important property to consider are the asymptotes.
Vertical Asymptotes
Almost all graphs of rational functions have vertical asymptotes, i.e. asymptotes with equations of the form x = a, where a is a constant. To find the vertical asymptote, set g(x) = 0 and solve for x.
Example: In the graph of y = 1/x, x = 0 is the vertical asymptote.
Horizontal Asymptotes
Some graphs of rational functions have horizontal asymptotes. These are asymptotes with equations of the form y = a, where a is a constant. There are two categories of horizontal asymptotes.
When degree of numerator < degree of denominator
When the degree of f(x) is less than that of g(x), the horizontal asymptote is always y = 0. This can be verified by finding the limit as x tends to infinity. You'll find that y will tend to zero.
When degree of numerator = degree of denominator
When the degree of f(x) = degree of g(x), the horizontal asymptote is found by splitting the function y into its partial fractions:
y = c + fraction
The horizontal asymptote will be y = c, where c is a constant.
Oblique Asymptotes
Oblique asymptotes occur when the degree of the numerator is greater than that of the degree of the denominator. It can be linear, quadratic, cubic, quartic and so on. Here we focus on the linear oblique asymptote, but those of curves can be found similarly.
Similar to the horizontal asymptote, first split y into its partial fractions.
y = mx + c + fraction
y = mx + c is the oblique asymptote.
Example: Sketch the graph of
Solution:
Split into partial fractions.
From here, we can tell that
Vertical asymptote: x = -1
Horizontal asymptote: y = 1
Intercepts:
when y = 0, x = 1
when x = 0, y = -1
Turning point:
dy/dx = 0
---> [x + 1 - (x - 1)] / (x + 1)2 = 0
---> 2 / (x + 1)2 = 0 --> no solution
There are no turning points.
Gradient:
dy/dx is always positive.
The gradient is always increasing.
Actually, this example is exactly the same as the one we used in our section Rectangular Hyperbolas. Hence, you can see the two methods that can be used for deriving the shape of this graph. Obviously, the method described in this page is more general as it can apply to graphs of any rational function, unlike the other method described earlier.
Below is another example to illustrate what we mean.
Example: Sketch the graph of
Solution:
Express as partial fractions.
From here, we can derive the asymptotes--
Vertical: x = 2
Oblique : y = 2x
Intercepts:
when x = 0, y = -3/2
when y = 0, x = 2x2 - 2x + 3
Discriminant = -20 < 0 --> no real roots.
There are no x-intercepts.
Turning points:
Find the corresponding y-values.
Curve Sketching | Quadratic Equation | Rectangular Hyperbolas | Conic sections | Modulus Functions
