To graph the functions given, we will primarily focus on the parent function \(f(x) = x^2\) and its transformation \(-f(x) = -x^2\). Here’s how you can represent these functions graphically.
Step-by-Step Instructions
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Set Up the Axes:
- Draw a vertical axis (y-axis) and a horizontal axis (x-axis).
- Mark the axes with appropriate scale (e.g., -4 to 4 for both axes) to capture the key points of the functions.
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Graph the Parent Function \(f(x) = x^2\):
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Calculate a few points:
- \(f(-2) = (-2)^2 = 4\)
- \(f(-1) = (-1)^2 = 1\)
- \(f(0) = 0^2 = 0\)
- \(f(1) = (1)^2 = 1\)
- \(f(2) = (2)^2 = 4\)
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Plot the points: (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4).
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Draw a smooth curve through these points to complete the parabola opening upwards.
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Graph the Transformed Function \(-f(x) = -x^2\):
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Calculate a few points:
- \(-f(-2) = -(-2)^2 = -4\)
- \(-f(-1) = -(-1)^2 = -1\)
- \(-f(0) = -0^2 = 0\)
- \(-f(1) = -(1)^2 = -1\)
- \(-f(2) = -(2)^2 = -4\)
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Plot the points: (-2, -4), (-1, -1), (0, 0), (1, -1), (2, -4).
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Draw a smooth curve through these points to complete the parabola opening downwards.
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Label the Graphs:
- Label the curve for \(f(x) = x^2\) as “\(y = x^2\)” in a color such as blue.
- Label the curve for \(-f(x) = -x^2\) as “\(y = -x^2\)” in a different color, such as red.
Summary of Graph Features
- The function \(f(x) = x^2\) is a parabola that opens upward, with its vertex at the origin (0,0) and symmetric about the y-axis.
- The function \(-f(x) = -x^2\) is a parabola that opens downward, also with its vertex at the origin (0,0) and symmetric about the y-axis.
By following these steps, you will have a clear graphical representation of both the parent function and its negative transformation.
Note:
If you’re using graphing software or a graphing calculator, you can simply input these functions directly and adjust the window to properly view both curves, ensuring they are labeled appropriately for clarity.
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