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Ch. 1 - Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 1 - Functions문제 1.2.32
1장, 문제 1.2.32

Shifting Graphs


Exercises 27–36 tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation.


y = −√x Right 3

검증된 단계별 안내
1
Identify the transformation needed: The problem states that the graph of the equation y = -√x needs to be shifted to the right by 3 units.
Understand the effect of shifting: Shifting a graph to the right by 'h' units involves replacing 'x' with '(x - h)' in the equation. Here, h = 3.
Apply the transformation: Replace 'x' with '(x - 3)' in the original equation y = -√x to get the new equation y = -√(x - 3).
Sketch the graphs: Draw the original graph of y = -√x, which is a downward-opening half parabola starting at the origin (0,0). Then, draw the shifted graph of y = -√(x - 3), which starts at the point (3,0) and follows the same shape.
Label the graphs: Clearly label the original graph with the equation y = -√x and the shifted graph with the equation y = -√(x - 3) to distinguish between them.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
8m
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주요 개념

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

Graph Shifting

Graph shifting involves moving a graph horizontally or vertically without changing its shape. A horizontal shift occurs when a constant is added or subtracted from the x-variable, moving the graph left or right. For example, y = f(x - 3) shifts the graph of y = f(x) three units to the right.
추천 영상:
가이드 코스
5:53
Graph of Sine and Cosine Function

Square Root Function

The square root function, y = √x, is a basic function in calculus that produces a curve starting at the origin and increasing slowly. It is defined only for x ≥ 0, as the square root of a negative number is not real. Understanding its shape and domain is crucial for graph transformations.
추천 영상:
03:19
Completing the Square to Rewrite the Integrand Example 6

Equation Transformation

Transforming an equation involves modifying its algebraic expression to reflect changes in the graph. For a horizontal shift to the right by 3 units, replace x with (x - 3) in the equation. Thus, y = -√x becomes y = -√(x - 3), representing the shifted graph.
추천 영상:
가이드 코스
5:25
Intro to Transformations
관련 실천
교과서 질문

Even and Odd Functions


In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.


f(x) = x⁻⁵

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교과서 질문

In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.


𝔂 = x²/⁵

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교과서 질문

Use graphing software to graph the functions specified in Exercises 31–36.

Select a viewing window that reveals the key features of the function.


Graph the upper branch of the hyperbola y² − 16x² = 1.

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교과서 질문

In Exercises 35 and 36, find the (a) domain and (b) range.


𝔂 = { -x - 2, -2 ≤ x ≤ - 1

{ x, -1 < x ≤ 1

{ -x + 2, 1 < x ≤ 2

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교과서 질문

Theory and Examples


In Exercises 69 and 70, match each equation with its graph. Do not use a graphing device, and give reasons for your answer.


a. y = x⁴

b. y = x⁷

c. y = x¹⁰


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교과서 질문

Shifting Graphs


Exercises 27–36 tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation.


y = (1/2)(x + 1) + 5 Down 5, right 1

194
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