styles Vincent says that this is not possible in Asymptote, although he can see how it might be useful 2 Drawing a two-dimensional image 2 1 Lines and sizing Having determined that Asymptote is already installed on her computer, Janet decides to use it to draw a picture of the two-dimensional region that will be revolved
Oblique Asymptotes Each rational function of Example 2 had one horizontal asymptote and a vertical asymptote for each number that caused the denominator to be 0 The horizontal asymptote y 0 occurs because as x gets larger and larger, the y-coordinate gets closer and closer to 0 Some rational functions have a nonhorizontal line for an asymptote
Apr 13, 2011 · 4 If there is a horizontal asymptote y = b, determine whether the graph intersects this asymptote by solving f(x) = b If x can be solved, the intersection is located at this x-value If there is no solution for x, f(x) does not intersect the horizontal asymptote 5 Use the vertical asymptotes and x-intercepts to divide the x-axis into
the same asymptotes, domain, and range as the function f(x) = 1 — x STUDY TIP Notice that 1 — x → 0 as x → ∞ and as x → −∞ This explains why y = 0 is a horizontal asymptote of the graph of f(x) = 1 — You x can also analyze y-values as x approaches 0 to see why x = 0 is a vertical asymptote x y 4 2 2 4 f(x) = 1 x horizontal
asymptote Ask, “Why is there a gap at? Does this change the domain or the range?” Have students write the range and check it with a partner Discuss why there might be a gap at Set the function equal to 1 and have students try to solve the equation (While asymptote lines may be covered before or after this lesson, this is a good
-+- k Note that the asymptotes are affected only by the parameters h and k, while the reference points are affected by all four parameters Feature Vertical asymptote Horizontal asymptote Reference point Reference point f(x) g(x) a Example Identify the transformations of the graph of f (x) = — that produce the graph of the given function g(x)
Section 3 4 Rational Functions In the last few sections, we have built polynomials based on the positive whole number power functions In this section we explore functions based on power functions with negative integer powers, called rational functions Example 1 You plan to drive 100 miles Find a formula for the time the trip will take as a
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