교과서 질문
Solving Trigonometric Equations
For Exercises 51–54, solve for the angle θ, where 0 ≤ θ ≤ 2π.
sin² θ = cos² θ
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Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
1장, 문제 1.2.29
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 = x³ Left 1, down 1
검증된 단계별 안내1
Identify the original function: The given function is \( y = x^3 \).
Determine the horizontal shift: A shift to the left by 1 unit means replacing \( x \) with \( x + 1 \) in the function. This gives us \( y = (x + 1)^3 \).
Determine the vertical shift: A shift down by 1 unit means subtracting 1 from the entire function. This modifies the equation to \( y = (x + 1)^3 - 1 \).
Write the equation for the shifted graph: The new equation after applying both shifts is \( y = (x + 1)^3 - 1 \).
Sketch the graphs: Draw the original graph of \( y = x^3 \) and the shifted graph \( y = (x + 1)^3 - 1 \) on the same set of axes, labeling each graph with its respective equation.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Graph Shifting
Graph shifting involves translating a graph horizontally and vertically on the coordinate plane. A horizontal shift moves the graph left or right, while a vertical shift moves it up or down. For the equation y = x³, shifting left by 1 unit and down by 1 unit results in the new equation y = (x + 1)³ - 1.
추천 영상:
가이드 코스
5:53
Graph of Sine and Cosine Function
Horizontal Shift
A horizontal shift changes the x-values of a function. If a graph is shifted left by 'a' units, the transformation is represented by replacing x with (x + a) in the equation. For y = x³, shifting left by 1 unit modifies the equation to y = (x + 1)³, effectively moving the graph one unit to the left.
추천 영상:
가이드 코스
5:25Intro to Transformations
Vertical Shift
A vertical shift affects the y-values of a function. Shifting a graph down by 'b' units involves subtracting 'b' from the entire function. In the case of y = x³, shifting down by 1 unit results in the equation y = x³ - 1, which lowers the graph by one unit on the y-axis.
추천 영상:
가이드 코스
5:25Intro 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.
sin x²
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교과서 질문
Finding Formulas for Functions
Consider the point (x,y) lying on the graph of y = √(x − 3). Let L be the distance between the points (x,y) and (4,0). Write L as a function of y.
356
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교과서 질문
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = x cos x
240
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교과서 질문
Finding Formulas for Functions
Express the side length of a square as a function of the length d of the square’s diagonal. Then express the area as a function of the diagonal length.
534
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교과서 질문
General Sine Curves
For
f(x) = A sin ((2π/B)(x – C) +D
identify A, B, C, and D for the sine functions in Exercises 67–70 and sketch their graphs.
y = ½ sin (πx – x) + ½
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