Use shifts and scalings to transform the graph of ƒ(x)=x2ƒ(x)=$$x^2$$ into the graph of g. Use a graphing utility to check your work.g(x)=ƒ(2x−4)g(x) = ƒ(2x - 4)

Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 1 - Functions

문제 55b

1장, 문제 55b

Use shifts and scalings to transform the graph of $ƒ (x) = x^{2}$ into the graph of g. Use a graphing utility to check your work.

$g (x) = ƒ (2 x - 4)$

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Start with the function $ f(x) = x^2 $. This is a basic parabola opening upwards with its vertex at the origin (0,0).

Identify the transformation needed to obtain $ g(x) = f(2x - 4) $. This involves a horizontal scaling and a horizontal shift.

First, consider the expression $ 2x - 4 $. Factor out the 2 to get $ 2(x - 2) $. This indicates a horizontal compression by a factor of 2 and a shift to the right by 2 units.

Apply the horizontal compression: Replace $ x $ with $ 2x $ in $ f(x) $ to get $ f(2x) = (2x)^2 = 4x^2 $. This compresses the graph horizontally by a factor of 2.

Apply the horizontal shift: Replace $ x $ with $ x - 2 $ in $ f(2x) $ to get $ f(2(x - 2)) = 4(x - 2)^2 $. This shifts the graph 2 units to the right.

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주요 개념

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

Function Transformation

Function transformation involves altering the graph of a function through shifts, stretches, and reflections. In this case, the function f(x) = x² is transformed into g(x) = f(2x - 4) by applying horizontal and vertical shifts and scalings. Understanding how these transformations affect the graph is crucial for accurately sketching the new function.

Horizontal Shifts

Horizontal shifts occur when the input of a function is adjusted by adding or subtracting a constant. For g(x) = f(2x - 4), the term '−4' indicates a shift to the right by 4 units. This concept is essential for determining how the graph of the original function moves along the x-axis.

Vertical Scaling

Vertical scaling refers to stretching or compressing the graph of a function by multiplying the input by a constant factor. In g(x) = f(2x - 4), the '2' in front of x indicates a horizontal compression by a factor of 1/2. This concept helps in understanding how the shape of the graph changes in relation to the original function.

Use shifts and scalings to transform the graph of ƒ(x)=x2ƒ(x)=x^2 into the graph of g. Use a graphing utility to check your work.g(x)=ƒ(2x−4)g(x) = ƒ(2x - 4)