Rational functions are mathematical expressions defined as the quotient of two polynomials. This text delves into their properties, including asymptotes, which are lines the graph approaches but never touches, and intercepts, which are points where the graph crosses the axes. Understanding these elements is crucial for graphing rational functions, and the text provides a step-by-step approach to accurately depict their behavior.
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Rational functions are defined as the quotient of two polynomial functions
Definition of Polynomial Functions
Polynomial functions are expressions comprising variables with non-negative integer exponents and constant coefficients
Example of a Polynomial Function
An example of a polynomial function is \( y = x^5 + 6x^3 + 2x + 9 \)
The domain of a rational function excludes values that would make the denominator zero, leading to discontinuities or undefined points in the function
Definition of Asymptotes
Asymptotes are lines that the graph of a rational function approaches but does not intersect
Types of Asymptotes
Asymptotes can be vertical, horizontal, or slant (oblique)
Vertical asymptotes occur at values of \( x \) that make the denominator \( Q(x) \) zero, indicating where the function's value becomes unbounded
Determining Horizontal Asymptotes
Horizontal asymptotes are determined by the degrees of the numerator and denominator polynomials
Cases of Horizontal Asymptotes
Depending on the degrees of the numerator and denominator polynomials, a rational function may have a horizontal asymptote at the ratio of the leading coefficients, at \( y = 0 \), or may not have a horizontal asymptote
If the degree of the numerator is higher than that of the denominator, the function may approach infinity or negative infinity, depending on the leading coefficients and the highest degree terms
Definition of Intercepts
Intercepts are points where a graph crosses the axes
Finding Intercepts
The x-intercepts, or roots, are found by setting the numerator \( P(x) \) to zero and solving for \( x \). The y-intercept is determined by evaluating the function at \( x = 0 \)
Plotting Asymptotes
Begin by plotting any vertical and horizontal asymptotes as dashed lines
Plotting Intercepts
Find and plot the x and y-intercepts
Choosing Additional Points
Select \( x \) values, calculate the corresponding \( y \) values, and plot them on the graph
Sketching the Curve
Use the asymptotes and intercepts as guides to sketch the curve, ensuring it approaches the asymptotes appropriately as it extends towards infinity in either direction
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Finding vertical asymptotes method
Set Q(x) = 0 and solve for x; exclude solutions from domain.
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Horizontal asymptote when degrees of P(x) and Q(x) are equal
Horizontal asymptote is y = (leading coefficient of P(x)) / (leading coefficient of Q(x)).
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Horizontal asymptote when degree of P(x) < degree of Q(x)
Horizontal asymptote is at y = 0.
