Provide three characteristics of the function y=x that remain unchanged under the following transformations: a vertical stretch by a factor of 3 and a horizontal compression by a factor of 3.
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To determine the characteristics of the function y = x that remain unchanged under a vertical stretch by a factor of 3 and a horizontal compression by a factor of 3, let's break it down step-by-step:
1. Vertical stretch by a factor of 3:
To vertically stretch the function y = x by a factor of 3, we multiply the y-coordinate of each point on the graph by 3. This will elongate the graph vertically. The original function y = x becomes y = 3x.
2. Horizontal compression by a factor of 3:
To horizontally compress the function y = 3x by a factor of 3, we divide the x-coordinate of each point on the graph by 3. This will squeeze the graph horizontally. The function y = 3x becomes y = 3(x/3), which simplifies to y = x.
Now that we have the transformed equation, y = x, let's identify the three characteristics that remain unchanged:
1. Slope:
The slope of the original function y = x is 1, and after the given transformations, the slope remains unchanged at 1.
2. x-intercept:
The x-intercept of the original function y = x is (0, 0), and after the transformations, the x-intercept remains unchanged.
3. y-intercept:
The y-intercept of the original function y = x is also (0, 0). Since the transformations did not affect the y-intercept, it remains the same as well.
So, the three characteristics of the function y = x that remain unchanged under a vertical stretch by a factor of 3 and a horizontal compression by a factor of 3 are the slope (1), the x-intercept (0), and the y-intercept (0).