A rational function is a function that can be expressed as the ratio of two polynomials:
$$f(x) = \frac{p(x)}{q(x)}$$
where \$p(x)\$ and \$q(x)\$ are polynomials and \$q(x) \neq 0\$.
NoteThe domainof a rational function is all real numbers except where the denominator is zero.
Graphing Rational Functions
The graph of a rational function can have vertical and horizontal asymptotes.
Vertical Asymptotes
A vertical asymptote occurs at \$x = a\$ if the function approaches infinity or negative infinity as \$x\$ approaches \$a\$.
NoteVertical asymptotes occur where the denominator is zero and the numerator is not zero.
Horizontal Asymptotes
A horizontal asymptote occurs when the function approaches a constant value as \$x\$ approaches infinity or negative infinity.
TipUse the zoomand tracefeatures to explore the graph in more detail.
Solving Rational Equations
To solve a rational equation using a graphing calculator:
- Enter the left side of the equation as one function and the right side as another.
- Graph both functions on the same coordinate plane.
- Use the intersect feature to find the \$x\$-coordinate of the intersection point, which is the solution to the equation.
1. Graph the function \$f(x) = \frac{x^2 - 4}{x - 2}\$ on a graphing calculator. What do you observe at \$x = 2\$? 2. Solve the equation \$\frac{2x + 3}{x - 1} = 1\$ using a graphing calculator. What is the solution?
Theory of KnowledgeHow do graphing calculators change our understanding of functions and equations? Do they enhance or diminish our mathematical intuition?
