Explain how to use end behaviour to find the equation of the horizontal asymptote of a rational function.
just ignore everything but the highest power in the numerator and denominator
So, (3x^3-5x+2)/(2x^3+9x^2-100) looks like
3x^3/2x^2 = 3/2
for large x.
Or, you can think of it like this. Divide top and bottom by the highest power. That gives you
(3-5/x^2+2/x^3)/(2+9/x-100/x^3)
For large x, the fractions all go to zero, and we are just left with 3/2.
So, as x goes way out on the axis, y just approaches 3/2, the horizontal asymptote.
To find the equation of the horizontal asymptote of a rational function, we can use the concept of end behavior. The end behavior refers to how the function behaves as the input approaches positive infinity or negative infinity. By examining the degrees of the numerator and denominator of the rational function, we can determine the number of horizontal asymptotes and their equations.
Here's how you can use end behavior to find the equation of the horizontal asymptote of a rational function:
1. Determine the degrees of the numerator and denominator: Write the rational function in its simplified form, with the highest power of the variable in the numerator and denominator terms. For example, if the rational function is f(x) = (3x^2 + 2x - 1) / (2x + 1), the numerator has a degree of 2 (highest power of x) and the denominator has a degree of 1.
2. Compare the degrees: Compare the degrees of the numerator and denominator. There are three possible cases:
a. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0 (the x-axis). For example, f(x) = (2x + 1) / (3x^2 + 2x - 1) has a horizontal asymptote at y = 0.
b. If the degree of the numerator is equal to the degree of the denominator, then divide the coefficient of the leading term of the numerator by the coefficient of the leading term of the denominator. The result is the equation of the horizontal asymptote. For example, f(x) = (2x^2 + 3x - 1) / (3x^2 - 4x + 2) has a horizontal asymptote at y = 2/3.
c. If the degree of the numerator is greater than the degree of the denominator, it means that the rational function does not have a horizontal asymptote. Instead, it will have slant or oblique asymptotes.
By analyzing the end behavior and comparing the degrees of the numerator and denominator, we can determine the equation of the horizontal asymptote for a given rational function.