HPC Unit 2B Review

Front

What is a horizontal asymptote?

Back

A horizontal asymptote is a horizontal line that the graph of a function approaches as the input (x) approaches positive or negative infinity.

Front

How do you determine the horizontal asymptote of a rational function?

Back

For a rational function of the form \( \frac{P(x)}{Q(x)} \), compare the degrees of the polynomials P(x) and Q(x): 1. If degree of P < degree of Q, y = 0 is the horizontal asymptote. 2. If degree of P = degree of Q, y = \( \frac{a}{b} \) where a and b are the leading coefficients. 3. If degree of P > degree of Q, there is no horizontal asymptote.

Front

What is a hole in a graph?

Back

A hole in a graph occurs at a point where a function is not defined due to a common factor in the numerator and denominator of a rational function.

Front

How do you find the coordinates of a hole in a rational function?

Back

To find the coordinates of a hole, factor the numerator and denominator, identify the common factors, and set them equal to zero to find the x-coordinate. Substitute this x-value into the simplified function to find the y-coordinate.

Front

What is a slant asymptote?

Back

A slant (or oblique) asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator in a rational function.

Front

How do you find the slant asymptote of a rational function?

Back

To find the slant asymptote, perform polynomial long division on the rational function. The quotient (ignoring the remainder) will give the equation of the slant asymptote.

Front

What is the significance of the leading coefficient in determining horizontal asymptotes?

Back

The leading coefficient of the highest degree term in the numerator and denominator helps determine the horizontal asymptote when the degrees of the polynomials are equal.

Front

What does it mean if a function has no horizontal asymptote?

Back

If a function has no horizontal asymptote, it means that as x approaches positive or negative infinity, the function's values do not approach a finite number.

Front

What is the difference between a vertical asymptote and a hole?

Back

A vertical asymptote occurs at values of x that make the denominator zero but not the numerator, while a hole occurs at values that make both the numerator and denominator zero.

Front

How can you identify vertical asymptotes in a rational function?

Back

Vertical asymptotes can be identified by setting the denominator equal to zero and solving for x, provided that these x-values do not also make the numerator zero.