The end behavior of a function refers to how the function behaves as the input variable approaches positive or negative infinity. It describes the limiting values or patterns that the function exhibits as it extends towards the far left and right sides of its graph.
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The end behavior of a polynomial function is determined by the degree and sign of the leading coefficient.
Polynomial functions with an even degree and a positive leading coefficient will have a parabolic shape that opens upward, while those with a negative leading coefficient will open downward.
Polynomial functions with an odd degree will have a graph that approaches positive or negative infinity as the input variable approaches positive or negative infinity.
Rational functions can have horizontal, vertical, or oblique asymptotes, depending on the degrees of the numerator and denominator polynomials.
The end behavior of exponential and logarithmic functions is characterized by their rapid growth or decay as the input variable approaches positive or negative infinity.
Review Questions
Explain how the degree and sign of the leading coefficient of a polynomial function affect its end behavior.
The degree and sign of the leading coefficient of a polynomial function determine the overall shape and end behavior of the function's graph. Polynomial functions with an even degree and a positive leading coefficient will have a parabolic shape that opens upward, approaching positive infinity as the input variable approaches positive or negative infinity. Conversely, polynomial functions with an even degree and a negative leading coefficient will have a parabolic shape that opens downward, approaching negative infinity as the input variable approaches positive or negative infinity. Polynomial functions with an odd degree will have a graph that approaches positive infinity as the input variable approaches positive infinity and negative infinity as the input variable approaches negative infinity, regardless of the sign of the leading coefficient.
Describe the different types of asymptotes that can be exhibited by rational functions and how they are determined.
Rational functions can have three types of asymptotes: horizontal, vertical, and oblique. Horizontal asymptotes occur when the degree of the numerator polynomial is less than or equal to the degree of the denominator polynomial, and the ratio of the leading coefficients determines the horizontal asymptote's value. Vertical asymptotes occur at the values of the input variable where the denominator polynomial equals zero, indicating points where the function is undefined. Oblique asymptotes occur when the degree of the numerator polynomial is one more than the degree of the denominator polynomial, and the slope of the oblique asymptote is determined by the ratio of the leading coefficients.
Analyze how the end behavior of exponential and logarithmic functions differs from that of polynomial and rational functions.
The end behavior of exponential and logarithmic functions is fundamentally different from that of polynomial and rational functions. Exponential functions, which can be expressed in the form $a^x$, where $a > 0$, exhibit rapid growth or decay as the input variable approaches positive or negative infinity. As the input variable increases, the function approaches positive infinity, and as the input variable decreases, the function approaches zero. Logarithmic functions, which can be expressed in the form $\log_a(x)$, where $a > 0$ and $x > 0$, exhibit a different end behavior, approaching negative infinity as the input variable approaches zero from the positive side and positive infinity as the input variable approaches positive infinity. This contrasts with the parabolic or asymptotic end behavior of polynomial and rational functions.