Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

문제 53

3장, 문제 53

Begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. g(x) = x² - 2

검증된 단계별 안내

Start with the standard quadratic function f(x) = x². This is a parabola that opens upwards with its vertex at the origin (0, 0). The graph is symmetric about the y-axis.

Understand the transformation: The given function g(x) = x² - 2 is derived from f(x) = x² by subtracting 2. This represents a vertical shift downward by 2 units.

To graph g(x), take each point on the graph of f(x) = x² and move it 2 units downward. For example, the vertex (0, 0) of f(x) will move to (0, -2) for g(x).

Plot the new points after the vertical shift. For instance, if (1, 1) is a point on f(x), it will become (1, -1) on g(x). Similarly, (-1, 1) will become (-1, -1). Repeat this for other points.

Draw the new parabola for g(x) = x² - 2, ensuring it maintains the same shape as f(x) = x² but is shifted downward by 2 units. Label the vertex and key points to complete the graph.

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주요 개념

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

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax² + bx + c. The graph of a quadratic function is a parabola, which opens upwards if 'a' is positive and downwards if 'a' is negative. Understanding the basic shape and properties of quadratic functions is essential for analyzing transformations.

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a function. For quadratic functions, vertical shifts occur when a constant is added or subtracted from the function, such as in g(x) = x² - 2, which shifts the graph of f(x) = x² down by 2 units. Recognizing these transformations helps in accurately graphing modified functions.

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point on its graph, depending on its orientation. For the standard quadratic function f(x) = x², the vertex is at the origin (0,0). In the function g(x) = x² - 2, the vertex shifts to (0, -2), which is crucial for understanding the graph's position and shape after transformation.

관련 실천

교과서 질문

Use the graph of y = f(x) to graph each function g. g(x)=2f(x-1)

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교과서 질문

In Exercises 53–58, f and g are defined by the following tables. Use the tables to evaluate each composite function. f(g(1))

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교과서 질문

Find

a. (fog) (x)

b. (go f) (x)

c. (fog) (2)

d. (go f) (2).

f(x) = x+4, g(x) = 2x + 1

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교과서 질문

In Exercises 51–54, graph the given square root functions, f and g, in the same rectangular coordinate system. Use the integer values of x given to the right of each function to obtain ordered pairs. Because only nonnegative numbers have square roots that are real numbers, be sure that each graph appears only for values of x that cause the expression under the radical sign to be greater than or equal to zero. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = √x (x = 0, 1, 4, 9) and g(x) = √x −1 (x = 0, 1, 4, 9)

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교과서 질문

Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. x² + y²+6x+2y+6 = 0

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교과서 질문

In Exercises 39-52, a. Find an equation for ƒ¯¹(x). b. Graph ƒ and ƒ¯¹(x) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range off and ƒ¯¹. f(x) = ∛x + 1

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