Sure! To understand the transformations of the parent function \( f(x) = x^2 \) when we add or subtract a constant \( k \), we can analyze the following equations:
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The parent function: \[ y = f(x) = x^2 \]
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The transformed function with \( k = 4 \): \[ y = f(x) + k = x^2 + 4 \]
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For subtraction, it would be: \[ y = f(x) - k = x^2 - 4 \]
Graphing Instructions
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Parent Function \( y = x^2 \):
- This is a standard upward-opening parabola with its vertex at the origin (0,0).
- Important points to plot:
- (0,0)
- (-1, 1)
- (1, 1)
- (-2, 4)
- (2, 4)
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Transformed Function \( y = x^2 + 4 \):
- This graph is also a parabola that opens upwards but is shifted up by 4 units.
- Important points to plot:
- (0, 4)
- (-1, 5)
- (1, 5)
- (-2, 6)
- (2, 6)
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Transformed Function \( y = x^2 - 4 \):
- This graph is again a parabola that opens upwards but is shifted down by 4 units.
- Important points to plot:
- (0, -4)
- (-1, -3)
- (1, -3)
- (-2, 0)
- (2, 0)
Visual Representation
Here's a description of how to sketch it:
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Sketch the Parent Function:
- Draw a parabola opening upwards, passing through the origin and plotting the points mentioned above.
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Sketch \( y = x^2 + 4 \):
- Draw another upward-opening parabola parallel to the first, shifted 4 units upwards. Make sure to label this graph as \( y = x^2 + 4 \).
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Sketch \( y = x^2 - 4 \):
- Similarly, draw yet another upward-opening parabola parallel to the first but shifted 4 units downwards. Label this graph as \( y = x^2 - 4 \).
Labeling
Be sure to include labels next to each curve:
- For the parent function: "y = x^2"
- For the transformed function up: "y = x^2 + 4"
- For the transformed function down: "y = x^2 - 4"
Conclusion
You now have a clear visual representation of the parent function and its transformations by vertically shifting it up or down by 4 units. If you choose to draw or use graphing software, it will reinforce the concepts of transformations in a quadratic function.
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