8.5.5 Functions with Asymptotes and Critical Points

Question 1

Functions with Asymptotes and Critical Points

Answer 1

• 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.

Question 2

note

Answer 2

• 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.

Question 3

Choose the graph of the following equation.

f(x)=∣x^2∣+3 / x^2+1

Answer 3

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Question 4

Which of the following curves is the graph of the equation

f(x) = 3x^2+4x / 2x^2−1?

Answer 4

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Question 5

Which of the following curves is the graph of the equation

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

Answer 5

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