Understanding Asymptotes and Graphing Functions

The video tutorial explains how to determine and graph the asymptotes of a rational function. It starts by identifying vertical asymptotes at the zeroes of the denominator, provided they are not zeroes of the numerator. The vertical asymptote is found at x = 4. Next, the horizontal asymptote is determined by evaluating the limit of the function as x approaches infinity, resulting in y = 2. The tutorial then demonstrates graphing the function by plotting points using a T-table, starting with the origin and additional points like (2,-2), (6,6), and (8,4). The graph approaches the asymptotes, and the process is summarized as selecting convenient x-values to plot points and understanding the graph's behavior near asymptotes.

What is the condition for a vertical asymptote to exist?

• The function must be undefined.

• Both numerator and denominator must be zero.

• The denominator must be zero and not a zero of the numerator.

• The numerator must be zero.

How do you determine the horizontal asymptote of a function?

• By finding the zeroes of the numerator.

• By setting the denominator equal to zero.

• By comparing the degrees of the numerator and denominator.

• By finding the x-intercept.

What is the horizontal asymptote of the function if the leading coefficients are 2 and 1?

• y = -2

• y = 0

• y = 2

• y = 1

What is the significance of the point (0,0) in the graph?

• It is the vertical asymptote.

• It is a point of discontinuity.

• It is the horizontal asymptote.

• It is the origin and the x- and y-intercept.

What is the y-value when x = 2 for the given function?

• -2

• 2

• 4

• 0

Why is it important to find additional points like (2, -2) when graphing?

• To accurately sketch the function's path.

• To find the x-intercept.

• To determine the vertical asymptote.

• To determine the horizontal asymptote.

What is the y-value when x = 6 for the given function?

• 2

• 6

• -6

• 0

What is the y-value when x = 8 for the given function?

• -4

• 4

• 8

• 0

What is the purpose of selecting convenient x-values when graphing?

• To determine the function's domain.

• To accurately plot the function's path.

• To simplify the function.

• To find the vertical asymptote.

What does the graph of a rational function typically do near its asymptotes?

• It intersects them.

• It approaches them.

• It becomes undefined.

• It remains constant.