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Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Blitzer 8th EditionCh. 2 - Functions and Graphs문제 95a
3장, 문제 95a

Begin by graphing the standard cubic function, f(x) = x³. Then use transformations of this graph to graph the given function. g(x) = x³-3

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Start by identifying the standard cubic function, f(x) = x³. This is a basic cubic function with a graph that passes through the origin (0, 0) and has a characteristic S-shape, increasing to the right and decreasing to the left.
Understand the transformation applied to f(x) to obtain g(x). The given function is g(x) = x³ - 3. This represents a vertical shift of the graph of f(x) downward by 3 units.
To apply the vertical shift, take each point on the graph of f(x) = x³ and subtract 3 from its y-coordinate. For example, the point (0, 0) on f(x) becomes (0, -3) on g(x). Similarly, the point (1, 1) on f(x) becomes (1, -2) on g(x), and so on.
Sketch the transformed graph. The new graph will have the same S-shape as the original cubic function, but it will be shifted downward by 3 units. Ensure that the key points, such as the origin (now at (0, -3)), are clearly marked.
Verify the transformation by substituting specific x-values into g(x) = x³ - 3 and confirming that the resulting points match the shifted graph. For example, if x = 2, g(2) = 2³ - 3 = 8 - 3 = 5, so the point (2, 5) should lie on the graph.

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주요 개념

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

Cubic Functions

A cubic function is a polynomial function of degree three, typically expressed in the form f(x) = ax³ + bx² + cx + d. The standard cubic function, f(x) = x³, has a characteristic S-shaped curve that passes through the origin and extends infinitely in both directions. Understanding the basic shape and properties of cubic functions is essential for analyzing transformations.
추천 영상:
4:56
Function Composition

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a function. For the function g(x) = x³ - 3, the transformation is a vertical shift downward by 3 units. Recognizing how these transformations affect the original graph is crucial for accurately sketching the new function.
추천 영상:
5:25
Intro to Transformations

Vertical Shifts

A vertical shift occurs when a constant is added to or subtracted from a function, resulting in the entire graph moving up or down. In the case of g(x) = x³ - 3, the '-3' indicates that every point on the graph of f(x) = x³ is lowered by 3 units. This concept is fundamental for understanding how to manipulate the graph of a function based on its equation.
추천 영상:
5:34
Shifts of Functions
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