Asymptotes, the lines that a graph aims for but never intersects, show how the graph behaves at its most extreme values. When the denominator of a function becomes zero, we see vertical asymptotes. If the denominator's degree surpasses that of the numerator, we encounter horizontal asymptotes. Lastly, when the degrees are identical, we have oblique asymptotes.
| Topic | Problem | Solution |
|---|---|---|
| None | Let $f(x)=\frac{3 x-6}{x^{2}+8 x+15}=\frac{(3 x-6… | First, we simplify the function $f(x)=\frac{3 x-6}{x^{2}+8 x+15}=\frac{3(x-2)}{(x+5)(x+3)}$ |
| None | Let $f(x)=\frac{3 x-6}{x^{2}+8 x+15}=\frac{(3 x-6… | Let $f(x)=\frac{3 x-6}{x^{2}+8 x+15}=\frac{(3 x-6)}{(x+5)(x+3)}$ |
| None | Find the equation of the vertical asymptote and t… | The vertical asymptote of a rational function is found by setting the denominator equal to zero and… |
| None | 2. A rational function has the form $f(x)=\frac{a… | A vertical asymptote occurs when the denominator of a rational function is equal to zero. In this c… |
| None | Listen Let $y=f(x)=\frac{x^{4}+4 x^{3}+2 x^{2}}{x… | Find the zeros of the denominator: \(x^3 + x = x(x^2 + 1) = 0\) |
| None | Suppose that $f(x)=\frac{x-4}{(x-6)(x+8)}$ a. Wha… | The vertical intercept of a function is the point where the graph of the function intersects the y-… |
| None | Which of the following rational functions behaves… | The behavior of a rational function as \(x \rightarrow \pm \infty\) is determined by the degree of … |
| None | Show Intro/Instructions Determine the horizontal … | The horizontal asymptote of a function can be determined by looking at the degrees of the polynomia… |