Find the horizontal and vertical asymptotes of the curve.
y = 2x^2+x−1/x^2+x−42
x=? (smaller x-value)
x=? (larger x-value)
y=?
To find the horizontal and vertical asymptotes of the given curve, we need to analyze the behavior of the function as x approaches infinity or negative infinity.
First, let's find the horizontal asymptote. The horizontal asymptote is the value that the function approaches as x goes to infinity or negative infinity.
To find the horizontal asymptote, we look at the degrees of the numerator and denominator functions. In this case, the degree of the numerator is 2 and the degree of the denominator is also 2.
When the degrees of the numerator and denominator are the same, we can compare the leading coefficients to determine the horizontal asymptote. The leading coefficient is the coefficient of the highest power of x.
In this case, the leading coefficient of the numerator is 2 and the leading coefficient of the denominator is 1. Since the leading coefficients are the same, the horizontal asymptote is the ratio of their coefficients, which is 2/1 = 2.
So, the horizontal asymptote of the curve is y = 2.
Next, let's find the vertical asymptotes. Vertical asymptotes occur when the denominator of the fraction becomes zero, resulting in an undefined value for the function.
To find the vertical asymptotes, we need to solve the equation x^2 + x - 42 = 0 for x.
Factoring the quadratic equation, we have:
(x + 7)(x - 6) = 0
Setting each factor equal to zero, we get:
x + 7 = 0 or x - 6 = 0
Solving these equations, we find:
x = -7 or x = 6
Therefore, the vertical asymptotes of the curve are x = -7 and x = 6.
For the smaller x-value, we have x = -7, and for the larger x-value, we have x = 6.
To find the y-value, we substitute these x-values into the equation:
For x = -7:
y = 2(-7)^2 + (-7) - 1 / (-7)^2 + (-7) - 42
y = 98 - 7 - 1 / 49 - 7 - 42
y = 90 / 0 (division by zero is undefined)
Since division by zero is undefined, there is no y-value for x = -7.
For x = 6:
y = 2(6)^2 + 6 - 1 / (6)^2 + 6 - 42
y = 72 + 6 - 1 / 36 + 6 - 42
y = 77 / 0 (division by zero is undefined)
Similarly, there is no y-value for x = 6.
Therefore, there are no y-values for both x = -7 and x = 6 as they correspond to the vertical asymptotes.