Maths-

General

Easy

Question

# What are the vertical and horizontal asymptotes of the graph of each 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= -3 and x = -4. and horizontal asymptote is y = (leading coefficient of numerator) / (leading coefficient of denominator) = 2/1 =2

- Find the asymptotes of the rational function, if any.
- Draw the asymptotes as dotted lines.
- Find the x -intercept (s) and y -intercept of the rational function, if any.
- Find the values of y for several different values of x .
- 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 .

x^{2} - 2x - 8 = 0

x^{2} + 2x - 4x - 8 = 0

x(x + 2)- 4(x + 2) = 0

(x + 2)(x - 4 )= 0

x = -2 or x = 4

The vertical asymptote of the rational function is x =−2 and x = 4

This function has x -intercept at (-2.386,0) and y -intercept at (0,1.125) . 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= -3 and x = -4. and horizontal asymptote is

y = (leading coefficient of numerator) / (leading coefficient of denominator) = =2

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