In Exercises 67–69, begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. r(x) = (1/2) |x + 2|

검증된 단계별 안내

Start with the basic absolute value function \(f(x) = |x|\), which is graphed as a V-shaped graph with its vertex at the origin (0,0).

Identify the transformation inside the absolute value: \(x + 2\). This represents a horizontal shift of the graph. Since it is \(x + 2\), shift the graph 2 units to the left.

Next, look at the coefficient outside the absolute value, which is \(\frac{1}{2}\). This is a vertical compression by a factor of \(\frac{1}{2}\), meaning the graph will be 'flatter' compared to the original \(f(x) = |x|\) graph.

Apply the horizontal shift first: move the vertex from (0,0) to (-2,0).

Then apply the vertical compression: multiply all the y-values of the shifted graph by \(\frac{1}{2}\). This will give you the graph of \(r(x) = \frac{1}{2} |x + 2|\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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주요 개념

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

Absolute Value Function

The absolute value function, f(x) = |x|, outputs the distance of x from zero on the number line, always producing non-negative values. Its graph is a V-shaped figure with the vertex at the origin (0,0), where the function changes direction. Understanding this basic shape is essential for applying transformations.

Horizontal Shifts

A horizontal shift moves the graph left or right without changing its shape. For r(x) = (1/2)|x + 2|, the '+2' inside the absolute value shifts the graph 2 units to the left. This means the vertex of the graph moves from (0,0) to (-2,0).

Vertical Scaling

Vertical scaling changes the steepness of the graph by multiplying the function by a constant. In r(x) = (1/2)|x + 2|, the factor 1/2 compresses the graph vertically, making it less steep than the original absolute value function. This affects the slope of the lines forming the V shape.