8.5.4 Functions with Asymptotes and Holes Flashcards
Functions with Asymptotes and Holes
• Identify vertical asymptotes for a rational function by factoring the numerator and denominator, canceling where possible, and determining where the resulting denominator is zero. A vertical asymptote to the graph of a function f is a line whose equation is x = a, where
, or .
• Identify horizontal asymptotes by taking the limit of the function as x approaches positive or negative infinity. A horizontal asymptote to the graph of a function f is a line whose equation is y = a, where , or .
• A hole (or point discontinuity) occurs in the graph of a function f at a point c if not equal to exists and f(c) is undefined or .
note
- When graphing a rational function, first look for
vertical asymptotes. - This function can be factored. The numerator and
denominator have a common factor of (x + 3), so cancel it. Make sure to promise not to evaluate the function at x = –3, because that would make the original expression undefined. - Now only x = 3 makes the denominator equal to zero, so that gives the location of the vertical asymptote.
- Notice that the simplified expression for the function
resembles a function you have already graphed. It has a horizontal asymptote at y = 1, but no critical point and no points of inflection. - The function is decreasing both to the left and right of x = 3. On the left it is concave down, and on the right it is concave up.
- Notice that there is a hole at x = –3. Since the function cannot be evaluated at this point, the graph skips over it, as indicated by the open circle. Since the function is not continuous at that point, it is also called a point discontinuity.
Which of the following curves is the graph of the equation
f(x) = x+3 / x^2+4x+3?
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Which of the following curves is the graph of the equation
f(x) = x^2−x−6 / x^2+x−2?
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Which of the following curves is the graph of the equation
f(x) = −x^2+x+6 / x^2+x−2?
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