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Maths-

General

Easy

Question

Asthon said the graph of $f left parenthesis x right parenthesis equals fraction numerator x plus 2 over denominator 2 x squared plus 4 x minus 6 end fraction$ has a horizontal asymptote at $y equals 0.5$ . Describe and correct Asthon's error.

hint 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) = $fraction numerator straight p left parenthesis straight x right parenthesis over denominator straight q left parenthesis straight x right parenthesis end fraction$ , 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: The vertical asymptote of the rational function is x= 2.24 and x= -2.24 . We will find more points on the function and graph the function.

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 .
2x² + 4x - 6= 0
x² + 2x - 3= 0
x² + 3x - x - 3 = 0
x(x + 3) - (x - 3) = 0
(x - 1) (x + 3) = 0
x = -3 or x = $2 over 2$ = 1
The vertical asymptote of the rational function is x= 2.24 and x= -2.24 .
We will find more points on the function and graph the function.

the degree of the numerator is less than the denominator, then the horizontal asymptote is located at y = 0 (which is the x-axis).

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