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AP Calculus AB: 8.5.5 Functions with Asymptotes and Critical Points

Mathematics5 CardsCreated 3 months ago

This content describes how to analyze rational functions with vertical and horizontal asymptotes, along with identifying critical points and inflection points using derivatives. It explains how asymptotes divide the graph into intervals with distinct behaviors in terms of increasing/decreasing and concavity.

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Functions with Asymptotes and Critical Points

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 .
• The behavior of a function can change from one side of a vertical asymptote to the other.

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Key Terms

Term
Definition

Functions with Asymptotes and Critical Points

Identify vertical asymptotes for a rational function by factoring the numerator and denominator, canceling where possible, and determining where th...

note

  • When graphing a rational function, first look for
    vertical asymptotes and horizontal asymptotes.

  • The denominator of this ...

Choose the graph of the following equation.

f(x)=∣x^2∣+3 / x^2+1

The graph of f(x) = (|x²| + 3) / (x² + 1) is symmetric about the y-axis, has a minimum at (0, 3), and approaches the horizontal asymptote y = 1 as ...

Which of the following curves is the graph of the equation

f(x) = 3x^2+4x / 2x^2−1?

The graph of f(x) = (3x² + 4x) / (2x² − 1) has vertical asymptotes at x = ±√(1/2), a horizontal asymptote at y = 3/2, and behavior changing around ...

Which of the following curves is the graph of the equation

f(x) = |x| / 3x+1?

The graph of f(x) = |x| / (3x + 1) has a vertical asymptote at x = -1/3, is positive when x > 0, negative when x < 0, and approaches zero as ...

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AP Calculus AB: 8.5.5 Functions with Asymptotes and Critical Points
TermDefinition

Functions with Asymptotes and Critical Points

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 .
• The behavior of a function can change from one side of a vertical asymptote to the other.

note

  • When graphing a rational function, first look for
    vertical asymptotes and horizontal asymptotes.

  • The denominator of this function can be factored, but nothing cancels. There are two vertical asymptotes.

  • Taking the limit of the function at infinity indicates that there is a horizontal asymptote at y = 0. Notice that the degree of the denominator is greater than that of the numerator.

  • The first derivative is never equal to 0. It is only undefined at the locations of the vertical asymptotes, which are indicated on the sign chart.

  • The second derivative equals 0 at x = 0. According to the sign chart, the concavity changes at that point, so it is a point of inflection.

  • Use the asymptotes and sign charts to graph the function. The vertical asymptotes partition the plane into three regions, and the function is decreasing on each region. On the leftmost region the function is concave down. In the middle region the function changes from concave up to concave down at x = 0. On the rightmost region the function is concave up.

Choose the graph of the following equation.

f(x)=∣x^2∣+3 / x^2+1

The graph of f(x) = (|x²| + 3) / (x² + 1) is symmetric about the y-axis, has a minimum at (0, 3), and approaches the horizontal asymptote y = 1 as x goes to infinity.

Which of the following curves is the graph of the equation

f(x) = 3x^2+4x / 2x^2−1?

The graph of f(x) = (3x² + 4x) / (2x² − 1) has vertical asymptotes at x = ±√(1/2), a horizontal asymptote at y = 3/2, and behavior changing around these asymptotes.

Which of the following curves is the graph of the equation

f(x) = |x| / 3x+1?

The graph of f(x) = |x| / (3x + 1) has a vertical asymptote at x = -1/3, is positive when x > 0, negative when x < 0, and approaches zero as x approaches infinity.