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

검증된 단계별 안내

Start by understanding the given function g(x) = -f(x + 1) - 1. This function is a transformation of the base function f(x). Each transformation will be applied step by step to the graph of f(x).

The first transformation is f(x + 1). This represents a horizontal shift of the graph of f(x) to the left by 1 unit. To apply this, take each point (x, y) on the graph of f(x) and move it to (x - 1, y).

Next, apply the negative sign in front of f(x), which is -f(x). This reflects the graph of f(x) across the x-axis. After the horizontal shift, take each point (x, y) and transform it to (x, -y).

Now, apply the vertical shift of -1. This means shifting the graph downward by 1 unit. After the reflection, take each point (x, y) and move it to (x, y - 1).

Finally, combine all the transformations to graph g(x). Start with the graph of f(x), shift it left by 1 unit, reflect it across the x-axis, and then shift it downward by 1 unit. The resulting graph is the graph of g(x).

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주요 개념

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Function Transformation

Function transformation refers to the changes made to the graph of a function based on modifications to its equation. These transformations include shifts, reflections, and stretches. In the given function g(x) = -f(x + 1) - 1, the graph of f(x) undergoes a horizontal shift to the left by 1 unit, a vertical reflection across the x-axis, and a downward shift by 1 unit.

Horizontal Shift

A horizontal shift occurs when the input of a function is altered, resulting in the entire graph moving left or right. In the function g(x) = -f(x + 1), the term (x + 1) indicates a shift to the left by 1 unit. This means that every point on the graph of f(x) will be moved leftward, affecting the x-coordinates of the graph.

Vertical Reflection and Shift

Vertical reflection and shift involve flipping the graph over the x-axis and moving it up or down. The negative sign in front of f(x) in g(x) = -f(x + 1) indicates a reflection across the x-axis, while the subtraction of 1 signifies a downward shift of the entire graph by 1 unit. This alters the y-coordinates of the points on the graph, creating a new transformed graph.