Find the horizontal and vertical asymptotes of the curve.
y = x^2+5/9x^2−44x−5
x= (smaller x-value)
x= (larger x-value)
y=
To find the horizontal and vertical asymptotes of the given curve, we need to examine the behavior of the function as x approaches positive or negative infinity.
1. Horizontal Asymptote:
To find the horizontal asymptote, we need to compare the degrees of the numerator and denominator. In this case, the degree of the numerator is 2, and the degree of the denominator is also 2.
When the degrees are the same, we divide the coefficient of the highest power term in the numerator by the coefficient of the highest power term in the denominator.
In this case, the coefficient of the highest power term in the numerator is 1, and the coefficient of the highest power term in the denominator is 9.
Thus, the horizontal asymptote is y = 1/9.
2. Vertical Asymptotes:
To find the vertical asymptotes, we need to find the values of x that make the denominator equal to zero.
Setting the denominator, 9x^2 - 44x - 5, equal to zero and solving for x, we get:
9x^2 - 44x - 5 = 0
To factor this expression or solve for x using the quadratic formula, we need to determine if it is factorable. Checking the discriminant:
b^2 - 4ac = (-44)^2 - 4(9)(-5) = 1936 + 180 = 2116
Since the discriminant is positive, the quadratic expression is factorable.
Factoring the expression, we have:
(3x + 1)(3x - 5) = 0
Setting each factor equal to zero and solving for x:
3x + 1 = 0 ⟹ 3x = -1 ⟹ x = -1/3
3x - 5 = 0 ⟹ 3x = 5 ⟹ x = 5/3
Therefore, the vertical asymptotes occur at x = -1/3 and x = 5/3.
In summary, the horizontal asymptote is y = 1/9, and the vertical asymptotes occur at x = -1/3 and x = 5/3.
To find the horizontal asymptote of a function, we need to determine the behavior of the function as x approaches positive infinity (∞) and negative infinity (-∞).
For this function, y = (x^2 + 5)/(9x^2 - 44x - 5), we compare the degrees of the numerator and denominator.
The numerator has the highest degree of 2, and the denominator also has the highest degree of 2. When the degrees of the numerator and denominator are the same, we can find the horizontal asymptote by comparing the coefficients of the highest degree terms.
In this case, the highest degree term in both the numerator and denominator is x^2. Thus, the horizontal asymptote can be determined by dividing the coefficients of the highest degree terms. The horizontal asymptote (HA) is given by:
HA = coefficient of the highest degree term in the numerator / coefficient of the highest degree term in the denominator.
Coefficient of x^2 in numerator = 1
Coefficient of x^2 in denominator = 9
HA = 1/9
So, the horizontal asymptote of the curve is y = 1/9.
Now let's find the vertical asymptotes.
To find the vertical asymptotes, we need to check if there are any values of x that make the denominator equal to zero, resulting in an undefined value (division by zero).
To find these values, set the denominator equal to zero and solve for x:
9x^2 - 44x - 5 = 0
Unfortunately, this equation cannot be easily factored. So, to find the values of x, we can use the quadratic formula:
x = [-b ± √(b^2 - 4ac)] / 2a
For our equation, a = 9, b = -44, and c = -5.
x = [-(-44) ± √((-44)^2 - 4(9)(-5))] / (2 * 9)
Simplifying, we get:
x = (44 ± √(1936 + 180)) / 18
x = (44 ± √2116) / 18
x = (44 ± 46) / 18
x = 90/18 = 5 or x = -2/9
So, the vertical asymptotes of the curve are x = 5 and x = -2/9.
To summarize:
Horizontal asymptote: y = 1/9
Vertical asymptotes: x = 5 and x = -2/9
for the horizontal asymptote, as x gets large, only the highest powers matter. So, what is x^2/9x^2 ?
For vertical asymptotes, they can only occur where the denominator is zero.
9x^2-44x-5 = (9x+1)(x-5)
so, when is that zero?