First we examined when a=1, b=1, and c=1
Immediate observations:
There are two branches: the result of a rational function.
The y-axis is a vertical asymptote which means that as x approaches 0, the function is undefined.
There is a slant asymptote of y = x which means the degree of the polynomial of the numerator is greater than the ratio of the degree of the denominator. Furthermore, the ratio of the leading coefficients of the second degree term to the first degree term is 1.
Observations:
"a" controls the width of the branches and determines the types and location of the asymptotes
Click here for a movie on how "a" affects the graph.
"b" controls the vertical shift of the function
Click here for a movie of how "b" affects the graph.
Branches occupy Quadrants 1 and 3
Vertical Asymptote x = 0
Slant Asymptote y = x + 1
Branches occupy Quadrants !,4 and 2,3
Vertical asymptote x=0
Slant asymptote y = x + 1
The vertical and slant asymptotes do not change, but the branches' direction of opening does.
Click here for a movie of how "c' affects the graph
Rational Functions can be defined as the quotient of two polynomials. It has the form f(x) = g(x) / h(x),
where h(x) is not equal to zero. Therefore, what we started with was a rational function written in another form:
f(x)*h(x) = g(x), where h(x) is linear and g(x) is quadratic.
The branches of a rational function approach lines called asymptotes.
If the function is not defined when x = a, then the line with the equation x = a is a vertical asymptote
The line y = b is a horizontal asymptote for a function f(x) if f(x)---> b as x--->infinity or as x--->negative infinity.
The oblique line "l" is a slant asymptote for a function f(x) if the graph of f(x) approaches "l" as x---> infinity or as x---> negative infinity.
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