Find all horizontal and vertical asymptotes of f(x)= 2x^2 + 7x + 12/
2x^2 + 5x - 12
Do not omit parentheses in the numerator and denominator:
f(x)= (2x^2 + 7x + 12)/(2x^2 + 5x - 12)
The function is different without them.
All polynomials are continuous over ℝ.
For a rational function, where a polynomial is divided by another, vertical asymptotes occur at the zeroes of the denominator, or 2x²+5x-12 in the present case.
Factorize 2x²+5x-12 to get
(x+4)(2x-3)
Solve for
(x+4)=0 and (2x-3)=0 for the locations of vertical asymptotes.
Horizontal asymptotes are the limites of the function when x->-∞ or x->∞.
For the case of rational functions where the degree of the numerator equals that of the denominator, the limit reduces to the quotient of the leading terms, namely:
2x²/2x²=1
So the horizontal asymptote is at y=1.
To find the horizontal asymptote of the function f(x), we need to compare the degrees of the numerator and denominator polynomials.
Degree of the numerator = 2
Degree of the denominator = 2
Since the degrees are equal, we need to compare the coefficients of the highest degree terms in the numerator and denominator polynomials. In this case, the coefficients are both 2.
So, the horizontal asymptote of the function is given by the ratio of the coefficients of the highest degree terms. Therefore, the horizontal asymptote is y = 2/2, which simplifies to y = 1.
To find the vertical asymptotes, we need to look for values of x that would cause our denominator to be zero. We can do this by factoring the denominator polynomial.
2x^2 + 5x - 12 = (2x - 3)(x + 4)
To find the values of x for which the denominator becomes zero, we need to set each factor equal to zero:
2x - 3 = 0 --> x = 3/2
x + 4 = 0 --> x = -4
Therefore, the vertical asymptotes of the function are x = 3/2 and x = -4.
To find the horizontal and vertical asymptotes of the function, we need to analyze the behavior of the function as x approaches positive or negative infinity.
Horizontal Asymptote:
A horizontal asymptote is a horizontal line that the function approaches as x approaches positive or negative infinity.
To find the horizontal asymptote, we need to compare the degrees of the numerator and denominator of the function. In this case, both the numerator and denominator have a degree of 2.
When the degrees of the numerator and denominator are equal, we can determine the horizontal asymptotes by comparing the coefficients of the highest degree terms in the numerator and denominator.
The highest degree terms in the numerator and denominator of the given function are 2x^2 and 2x^2, respectively. So, the coefficients of these terms are 2 and 2.
Since the coefficients are equal, the horizontal asymptote of the function is a horizontal line given by y = (coefficient of highest degree term in the numerator) / (coefficient of highest degree term in the denominator).
Therefore, the horizontal asymptote is y = 2/2 = 1.
Vertical Asymptotes:
A vertical asymptote occurs when the denominator of the function becomes zero.
To find the vertical asymptotes, we need to solve the equation 2x^2 + 5x - 12 = 0 for x.
Factorizing the quadratic equation, we have:
(2x - 3)(x + 4) = 0
Setting each factor to zero, we get:
2x - 3 = 0 ---> 2x = 3 ---> x = 3/2
x + 4 = 0 ---> x = -4
So, there are two vertical asymptotes in this case, x = 3/2 and x = -4.
To summarize:
Horizontal Asymptote: y = 1
Vertical Asymptotes: x = 3/2 and x = -4