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General

Easy

Question

# Juanita is trying to determine the vertical and horizontal asymptotes for the graph of the function . Describe and correct the error Juanita made in determining the vertical and horizontal asymptotes.

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: x = -3 x = 4, y = 1 y = -4

### 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 .

x^{2} – x - 12= 0

x^{2} – 4x + 3x - 12= 0

x(x - 4) + 3(x - 4) = 0

(x - 4)(x + 3)

x = -3 and x = 4

The vertical asymptote of the rational function is x= -3 and x= 4

horizontal asymptote is

y = (leading coefficient of numerator) / (leading coefficient of denominator) = =1

Juanita has taken the horizontal asymptote as y = -4 is an error in her calculation.

^{2}– x - 12= 0

x

^{2}– 4x + 3x - 12= 0

x(x - 4) + 3(x - 4) = 0

(x - 4)(x + 3)

x = -3 and x = 4

The vertical asymptote of the rational function is x= -3 and x= 4

horizontal asymptote is

y = (leading coefficient of numerator) / (leading coefficient of denominator) = =1

Juanita has taken the horizontal asymptote as y = -4 is an error in her calculation.

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