Graphs of rational functions

Front

What is a rational function?

Back

Front

What are the vertical asymptotes of a rational function?

Back

Vertical asymptotes occur at the values of x that make the denominator Q(x) equal to zero, provided that these values do not also make the numerator P(x) equal to zero.

Front

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

Back

The horizontal asymptote can be found by comparing the degrees of the numerator and denominator: If the degree of P is less than the degree of Q, the asymptote is y=0; if they are equal, y=\frac{a}{b} where a and b are the leading coefficients.

Front

What is an x-intercept of a function?

Back

An x-intercept is a point where the graph of the function crosses the x-axis, which occurs when f(x) = 0.

Front

How do you determine the x-intercepts of a rational function?

Back

To find the x-intercepts, set the numerator P(x) equal to zero and solve for x.

Front

What is a slant (oblique) asymptote?

Back

A slant asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator, and it can be found using polynomial long division.

Front

What does the term 'asymptote' mean in the context of graphs?

Back

An asymptote is a line that the graph approaches but never touches or crosses, indicating the behavior of the function as x approaches certain values.

Front

What is the significance of the denominator in a rational function?

Back

The denominator determines the vertical asymptotes and the domain of the function, as it cannot be zero.

Front

How can you identify the correct graph of a rational function?

Back

To identify the correct graph, analyze the function's asymptotes, intercepts, and end behavior, and compare these features with the options provided.

Front

What is the domain of a rational function?

Back

The domain of a rational function consists of all real numbers except those that make the denominator zero.