Use transformations of f(x)=1/x or f(x)=1/x2 to graph each rational function. h(x)=(1/x) + 2

Ch. 3 - Polynomial and Rational Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

문제 47

4장, 문제 47

Use transformations of f(x)=1/x or f(x)=1/x² to graph each rational function. h(x)=(1/x) + 2

검증된 단계별 안내

Identify the base function given, which is \( f(x) = \frac{1}{x} \). This is a rational function with a vertical asymptote at \( x = 0 \) and a horizontal asymptote at \( y = 0 \).

Look at the given function \( h(x) = \frac{1}{x} + 2 \). Notice that it is the base function \( \frac{1}{x} \) plus 2, which means the graph of \( f(x) \) is shifted vertically.

Understand that adding 2 to the function shifts the entire graph up by 2 units. This means the horizontal asymptote, originally at \( y = 0 \), will move to \( y = 2 \).

The vertical asymptote remains unchanged at \( x = 0 \) because the transformation does not affect the denominator.

To sketch the graph, start with the graph of \( f(x) = \frac{1}{x} \), then shift every point up by 2 units, and draw the new horizontal asymptote at \( y = 2 \) while keeping the vertical asymptote at \( x = 0 \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Parent Rational Functions

The parent functions f(x) = 1/x and f(x) = 1/x² are basic rational functions with distinct shapes and asymptotes. Understanding their graphs, including vertical and horizontal asymptotes, is essential as they serve as the starting point for transformations.

Transformations of Functions

Transformations involve shifting, stretching, compressing, or reflecting the graph of a function. For h(x) = 1/x + 2, the '+ 2' indicates a vertical shift upward by 2 units, moving the entire graph and its horizontal asymptote accordingly.

Asymptotes of Rational Functions

Asymptotes are lines that the graph approaches but never touches. For rational functions like 1/x, vertical asymptotes occur where the denominator is zero, and horizontal asymptotes describe end behavior. Transformations affect the position of these asymptotes.