Question

Find a rational function ƒ having a graph with the given features. x-intercepts: (1, 0) and (3, 0) y-intercept: none vertical asymptotes: x=0 and x=2 horizontal asymptote: y=1

Accepted Answer

Identify the x-intercepts of the rational function. Since the function has x-intercepts at (1, 0) and (3, 0), the numerator of the rational function must have factors corresponding to these roots: $$(x - 1)$$ and $$(x - 3). $$Determine the vertical asymptotes. Vertical asymptotes occur where the denominator is zero but the numerator is not. Given vertical asymptotes at $$x = 0$$ and $$x = 2$$, the denominator must have factors $$(x)$$ and $$(x - 2). $$Write the general form of the rational function using the factors found: $$f(x) = \frac{a(x - 1)(x - 3)}{x(x - 2)}$$ where $$a is a $$constant to be determined. Use the horizontal asymptote to find the constant $$a. $$Since the horizontal asymptote is $$y = 1$$, compare the degrees of numerator and denominator (both degree 2) and set the ratio of the leading coefficients equal to 1. The leading coefficient of the numerator is $$a$$, and the denominator's leading coefficient is 1, so $$a = 1. $$Check the y-intercept by evaluating $$f(0). $$Since there is no y-intercept, $$f(0)$$ must be undefined (which it is, due to the vertical asymptote at $$x=0$$), confirming the function satisfies all given conditions.

Find a rational function ƒ having a graph with the given features.
x-intercepts: (1, 0) and (3, 0)
y-intercept: none
vertical asymptotes: x=0 and x=2
horizontal asymptote: y=1

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Rational Functions and Their Graphs

Intercepts of Rational Functions

Asymptotes of Rational Functions