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

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Step 1: Understand the given problem. The function g(x) = -f(x) + 1 is a transformation of the original function y = f(x). This involves two transformations: a reflection over the x-axis and a vertical shift.

Step 2: Apply the reflection transformation. The term -f(x) reflects the graph of y = f(x) over the x-axis. For every point (x, y) on the graph of y = f(x), the corresponding point on y = -f(x) will be (x, -y).

Step 3: Apply the vertical shift. The +1 in g(x) = -f(x) + 1 shifts the graph of y = -f(x) upward by 1 unit. For every point (x, -y) on y = -f(x), the corresponding point on g(x) will be (x, -y + 1).

Step 4: Combine the transformations. Start with the graph of y = f(x), reflect it over the x-axis to get y = -f(x), and then shift the entire graph upward by 1 unit to obtain g(x) = -f(x) + 1.

Step 5: Plot the transformed graph. Use the transformations to adjust key points from the original graph of y = f(x) and sketch the new graph of g(x). Ensure that the shape and behavior of the graph are consistent with the transformations applied.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Function Transformation

Function transformation refers to the process of altering the graph of a function through various operations, such as shifting, reflecting, or stretching. In this case, the function g(x) = -f(x) + 1 involves a reflection across the x-axis due to the negative sign and a vertical shift upwards by 1 unit.

Reflection Across the X-Axis

Reflection across the x-axis occurs when the output values of a function are negated. For the function g(x) = -f(x), every point on the graph of f(x) is mirrored over the x-axis, meaning if f(x) has a point (a, b), then g(x) will have the point (a, -b). This transformation changes the sign of the y-values.

Vertical Shift

A vertical shift involves moving the entire graph of a function up or down without altering its shape. In the function g(x) = -f(x) + 1, the '+1' indicates that after reflecting f(x) across the x-axis, the graph is then shifted upwards by 1 unit, resulting in a new set of y-values that are all increased by 1.

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