Question

# Find the vertical and horizontal asymptotes of rational function, then graph the function.

Hint:

### A rational function is a function that is the ratio of polynomials. Any function of one variable, x, is called a rational function if, it can be represented as f(x) = , where p(x) and q(x) are polynomials such that q(x) ≠ 0.

Rational functions are of the form y = f(x)y = fx , where f(x)fx is a rational expression .

- If both the polynomials have the same degree, divide the coefficients of the leading terms. This is your asymptote.
- If the degree of the numerator is less than the denominator, then the asymptote is located at y = 0 (which is the x-axis).
- If the degree of the numerator is greater than the denominator, then there is no horizontal asymptote.

## The correct answer is: From the graph we can analyze that the vertical asymptote of the rational function is x= -1/2 and horizontal asymptote is y = (leading coefficient of numerator) / (leading coefficient of denominator) = 1/2=0.5

### 1.Find the asymptotes of the rational function, if any.

2.Draw the asymptotes as dotted lines.

3.Find the x -intercept (s) and y -intercept of the rational function, if any.

4.Find the values of y for several different values of x .

5.Plot the points and draw a smooth curve to connect the points. Make sure that the graph does not cross the vertical asymptotes.

The vertical asymptote of a rational function is x -value where the denominator of the function is zero. Equate the denominator to zero and find the value of x .

2x + 1 = 0

x =

The vertical asymptote of the rational function is x =

We will find more points on the function and graph the function.

From the graph we can analyze that the vertical asymptote of the rational function is x= and horizontal asymptote is

y = (leading coefficient of numerator) / (leading coefficient of denominator) = =0.5

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### When will the graph of a rational function have no vertical asymptotes ? Give an example of such a function.

Let's say that r is a rational function.

Identify R's domain.

If necessary, reduce r(x) to its simplest form.

Find the x- and y-intercepts of the y=r(x) graph if one exists.

If the graph contains any vertical asymptotes or holes, locate where they are.

Then, identify and, if necessary, analyze r's behavior on each side of the vertical asymptotes.

Investigate R's final behavior. If one exists, locate the horizontal or slant asymptote.

The graph of y=r(x) can be drawn using a sign diagram and additional points if necessary.

### When will the graph of a rational function have no vertical asymptotes ? Give an example of such a function.

Let's say that r is a rational function.

Identify R's domain.

If necessary, reduce r(x) to its simplest form.

Find the x- and y-intercepts of the y=r(x) graph if one exists.

If the graph contains any vertical asymptotes or holes, locate where they are.

Then, identify and, if necessary, analyze r's behavior on each side of the vertical asymptotes.

Investigate R's final behavior. If one exists, locate the horizontal or slant asymptote.

The graph of y=r(x) can be drawn using a sign diagram and additional points if necessary.