Question

Solve each problem. Work each of the following. Sketch the graph of a function that does not intersect its horizontal asymptote y=1, has the line x=3 as a vertical asymptote, and has x-intercepts (2, 0) and (4, 0).

Accepted Answer

Understand the problem: We need to find a function with a horizontal asymptote at $$y=1$$, a vertical asymptote at $$x=3$$, and x-intercepts at $$(2,0)$$ and $$(4,0). $$Since the function has a vertical asymptote at $$x=3$$, the denominator of the function should have a factor of $$(x - 3)$$, which causes the function to be undefined at $$x=3. $$The x-intercepts at $$x=2$$ and $$x=4$$ mean the numerator of the function should have factors $$(x - 2)$$ and $$(x - 4)$$, so the function equals zero at these points. To have a horizontal asymptote at $$y=1$$, the degrees of the numerator and denominator should be the same, and the ratio of their leading coefficients should be 1. This suggests the numerator and denominator should be polynomials of the same degree with leading coefficients equal. Construct the function as $$f(x) = \frac{a(x - 2)(x - 4)}{a(x - 3)(x - b)}$$ where $$a is a $$constant and $$b is $$chosen so that the horizontal asymptote is $$y=1. $$Then simplify and verify the conditions, adjusting $$a$$ and $$b if $$necessary.

Solve each problem. Work each of the following. Sketch the graph of a function that does not intersect its horizontal asymptote y=1, has the line x=3 as a vertical asymptote, and has x-intercepts (2, 0) and (4, 0).

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Asymptotes of a Function

X-Intercepts of a Function

Graphing Rational Functions