Use shifts and scalings to transform the graph of ƒ(x)=xƒ(x)=\sqrt{x} into the graph of g. Use a graphing utility to check your work.g(x)=3x−1−5g(x)=3\sqrt{x-1}-5

Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 1 - Functions

문제 56d

1장, 문제 56d

Use shifts and scalings to transform the graph of $ƒ (x) = \sqrt{x}$ into the graph of g. Use a graphing utility to check your work.
$g (x) = 3 \sqrt{x - 1} - 5$

검증된 단계별 안내

Start with the base function $ f(x) = \sqrt{x} $.

Apply a horizontal shift to the right by 1 unit to get $ \sqrt{x-1} $. This accounts for the $ x-1 $ inside the square root.

Apply a vertical stretch by a factor of 3 to get $ 3\sqrt{x-1} $. This is done by multiplying the entire function by 3.

Apply a vertical shift downward by 5 units to get $ 3\sqrt{x-1} - 5 $. This is done by subtracting 5 from the entire function.

The transformed function is $ g(x) = 3\sqrt{x-1} - 5 $. Use a graphing utility to verify the transformation visually.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

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

Transformations of Functions

Transformations of functions involve shifting, stretching, compressing, or reflecting the graph of a function. For example, adding a constant to the input (x) shifts the graph horizontally, while adding a constant to the output (f(x)) shifts it vertically. Understanding these transformations is crucial for manipulating the graph of a function like f(x) = √x to achieve the desired form of g(x).

Vertical and Horizontal Shifts

Vertical shifts occur when a constant is added or subtracted from the function, affecting its position along the y-axis. Horizontal shifts are achieved by adding or subtracting a constant from the input variable, affecting the x-axis position. In the transformation from f(x) = √x to g(x) = 3√(x-1) - 5, the term (x-1) indicates a horizontal shift to the right by 1 unit, while the -5 indicates a vertical shift downward by 5 units.

Scaling and Stretching

Scaling refers to multiplying the function by a constant, which affects the steepness of the graph. A vertical scaling, such as multiplying by 3 in g(x) = 3√(x-1) - 5, stretches the graph away from the x-axis, making it steeper. Understanding how scaling impacts the graph is essential for accurately transforming f(x) into g(x) and predicting the resulting shape of the graph.

Use shifts and scalings to transform the graph of ƒ(x)=xƒ(x)=\(\sqrt{x}\) into the graph of g. Use a graphing utility to check your work.g(x)=3x−1−5g(x)=3\(\sqrt{x-1}\)-5