Graph each function over a two-period interval.
y = 1 + tan x

검증된 단계별 안내

Identify the function to be graphed: \(y = 1 + \tan x\). This is a vertical shift of the basic tangent function by 1 unit upwards.

Recall the period of the tangent function. The basic tangent function \(\tan x\) has a period of \(\pi\), so two periods correspond to an interval of length \(2\pi\).

Determine the interval over which to graph the function. For two periods, choose an interval such as \([-\pi, \pi]\) or \([0, 2\pi]\).

Note the vertical asymptotes of \(\tan x\), which occur where \(\cos x = 0\), i.e., at \(x = \frac{\pi}{2} + k\pi\) for any integer \(k\). These asymptotes will also be shifted vertically but remain at the same \(x\)-values.

Plot key points of \(\tan x\) within the chosen interval, then shift all \(y\)-values up by 1 to graph \(y = 1 + \tan x\). Mark the vertical asymptotes and sketch the curve approaching these asymptotes accordingly.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

도움이 되었나요?

주요 개념

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

Periodicity of the Tangent Function

The tangent function has a fundamental period of π, meaning its values repeat every π units. Understanding this periodicity is essential for graphing the function over a specified interval, such as two periods, which would be 2π for tangent.

Vertical Asymptotes of Tangent

Tangent has vertical asymptotes where the function is undefined, occurring at odd multiples of π/2. Recognizing these asymptotes helps in accurately sketching the graph, as the function approaches infinity near these points.

Vertical Translation of Functions

The function y = 1 + tan x represents a vertical shift of the basic tangent graph upward by 1 unit. Understanding vertical translations allows you to adjust the graph accordingly without altering its shape or period.

Graph each function over a two-period interval.y = 1 + tan x