To determine the equation of the cube root function graphed by Sam, you need to look for characteristics of the graph that would correspond to transformations of the parent function \(y = \sqrt[3]{x}\).
The parent function \(y = \sqrt[3]{x}\) can be transformed in several ways:
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Horizontal shift: To shift the graph left or right, you adjust the x-value inside the function.
- Left shift: \(y = \sqrt[3]{x + k}\)
- Right shift: \(y = \sqrt[3]{x - k}\)
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Vertical shift: You can shift the graph up or down by adding or subtracting a constant from the entire function.
- Up shift: \(y = \sqrt[3]{x} + k\)
- Down shift: \(y = \sqrt[3]{x} - k\)
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Vertical stretch/compression: You can multiply the function by a constant to stretch or compress it.
- Vertical stretch: \(y = a \cdot \sqrt[3]{x}\) where \(a > 1\)
- Vertical compression: \(y = a \cdot \sqrt[3]{x}\) where \(0 < a < 1\)
Based on the answer choices provided, we can analyze each equation, particularly looking for shifts and stretches.
- If the function has a transformation that shifts it down and to the right, it may involve a minus in the cube root expression and a minus outside for a vertical shift.
- If the function shifts up, it will have a positive constant added outside.
Now, looking at the options:
- \(y = 2\sqrt[3]{x-1} + 3\): This indicates a right shift by 1 unit and an upward shift by 3 units, with a vertical stretch by a factor of 2.
- \(y = 2\sqrt[3]{x+1} - 3\): This indicates a left shift by 1 unit and a downward shift by 3 units, with no vertical stretch.
- \(y = \sqrt[3]{x+1} - 3\): This indicates a left shift by 1 unit and a downward shift by 3 units without stretching.
- \(y = x - 1 + 3\): Seems incorrect as it doesn't represent a cube root function.
- \(y = \sqrt[3]{x - 1} + 3\): Indicates a right shift and upward shift, similar to the first.
- \(y = \sqrt[3]{x + 1} - 3\): Left shift and down shift.
From this analysis, if you were to describe the transformations based on a hypothetical graph, you would match the equation closest to what you see from the graph. Without the image, I cannot definitively choose a correct answer, but you should match the function with transformations you've noted.
If you see a cube root graph that matches a right shift and upward move or left and down, select the corresponding equation accordingly.
1 answer