The transformation you are describing involves the equation \( y = kf(x) \), where \( k \) is a constant factor. This type of transformation is known as vertical scaling or vertical stretching/compressing.
Understanding Vertical Scaling:
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Vertical Stretch:
- If \( k > 1 \), the function is stretched vertically. This means that the values of \( f(x) \) are multiplied by a factor greater than 1, making the graph taller. For example, if \( k = 2 \), the output values will be twice as large as those in the original function \( f(x) \).
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Vertical Compression:
- If \( 0 < k < 1 \), the function is compressed vertically. This means the values of \( f(x) \) are multiplied by a factor less than 1, making the graph shorter. For example, if \( k = 0.5 \), the output values will be half as large as those in the original function \( f(x) \).
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Reflection:
- If \( k < 0 \), the graph is both reflected over the x-axis and stretched or compressed depending on the absolute value of \( k \). For example, if \( k = -2 \), the graph is stretched vertically by a factor of 2 and flipped upside down.
Graphical Representation:
- When you graph \( y = f(x) \) and \( y = kf(x) \), you can visualize how the points on the graph of \( y = f(x) \) either move farther from the x-axis (if \( k > 1 \)) or closer to the x-axis (if \( 0 < k < 1 \)). The entire shape of the function will remain the same, but it will either grow taller or shorter based on the value of \( k \).
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