To understand the transformation that occurs when changing the equation from \( f(x) = -x \) to \( f(x) = -x + 4 \), we need to analyze the effects of the added constant.
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Starting Point: The graph of \( f(x) = -x \) is a straight line that has a slope of -1. This line decreases from left to right and passes through the origin (0, 0).
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Adding a Constant: When you modify the function to \( f(x) = -x + 4 \), you are adding a constant (4) to the original function \( f(x) = -x \).
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Transformation: The addition of the constant affects the vertical position of the graph. Specifically, adding 4 translates (or shifts) the graph of \( f(x) = -x \) vertically upward by 4 units.
Summary of the Transformation:
- The graph of \( f(x) = -x \) will shift up 4 units to create the graph of \( f(x) = -x + 4 \). The new line will still have the same slope of -1, but it will now cross the y-axis at (0, 4) instead of (0, 0).
So, every point on the original line gets moved up by 4 units. For example, the point (1, -1) on the original line \( f(x) = -x \) will move to (1, 3) on the new line \( f(x) = -x + 4 \).