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

검증된 단계별 안내

Step 1: Understand the transformations applied to the function f(x). The given function g(x) = -(1/2)f(x+2) involves three transformations: a horizontal shift, a vertical scaling, and a reflection.

Step 2: Start with the horizontal shift. The term (x+2) inside f(x) indicates a horizontal shift to the left by 2 units. This means every point on the graph of f(x) will move 2 units to the left.

Step 3: Apply the vertical scaling. The coefficient (1/2) in front of f(x+2) compresses the graph vertically by a factor of 1/2. This means the y-coordinates of all points on the graph will be halved.

Step 4: Apply the reflection. The negative sign in front of (1/2) reflects the graph across the x-axis. This means the y-coordinates of all points will be multiplied by -1, flipping the graph upside down.

Step 5: Combine all transformations. Start with the graph of f(x), shift it 2 units to the left, compress it vertically by a factor of 1/2, and then reflect it across the x-axis. The resulting graph is the graph of g(x).

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주요 개념

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

Function Transformation

Function transformation refers to the process of altering the graph of a function through various operations, such as shifting, stretching, or reflecting. In the given function g(x) = -(1/2)f(x+2), the transformations include a horizontal shift to the left by 2 units, a vertical compression by a factor of 1/2, and a reflection across the x-axis.

Horizontal Shifts

Horizontal shifts occur when a function is modified by adding or subtracting a value inside the function's argument. For g(x) = f(x+2), the '+2' indicates a shift to the left by 2 units. This means that every point on the graph of f(x) will move leftward, affecting the x-coordinates of the graph.

Vertical Compression and Reflection

Vertical compression and reflection involve scaling the output of a function and flipping it over the x-axis. In g(x) = -(1/2)f(x+2), the factor of -1 indicates a reflection across the x-axis, while the factor of 1/2 compresses the graph vertically, making it half as tall. This alters the y-values of the function, affecting the overall shape of the graph.