To determine the equation of the cube root function that Sam graphed, we need to analyze the properties of cube root functions and the information typically provided in an S-shaped curve.
A general cube root function is in the form of \(y = a\sqrt[3]{x-h} + k\), where:
- \( (h, k) \) is the translation of the function from the origin.
- \( a \) is a vertical stretch or compression factor.
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Graph Behavior: The cube root function crosses through the point \( (0, 0) \) for the standard form \( y = \sqrt[3]{x} \).
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Shifts: Depending on the values of \( h \) and \( k \), the graph will shift horizontally and vertically. Look for shifts in the provided options:
- The term \( x-1 \) indicates a shift right by 1.
- The term \( x+1 \) indicates a shift left by 1.
- The constant \( +3 \) indicates a shift upward by 3.
- The constant \( -3 \) indicates a shift downward by 3.
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Assess Options:
- Option 1: \(y = 2\sqrt[3]{x-1}+3\) (Right 1 and Up 3)
- Option 2: \(y = \sqrt[3]{x-1}+3\) (Right 1 and Up 3)
- Option 3: \(y = \sqrt[3]{x+1}-3\) (Left 1 and Down 3)
- Option 4: \(y = 2\sqrt[3]{x+1}-3\) (Left 1 and Down 3)
From this analysis, choose an option that suits typical characteristics of a cube root function. Without the exact details of the image content, if the S-shaped curve shows a broader curve or crosses particular points, we may prefer either option 1 or option 2 due to the positive vertical shift.
Given these considerations, the most likely candidates for a cube root function that has a vertical shift upwards are:
- \(y=\sqrt[3]{x-1}+3\)
So, if we assume a horizontal shift to the right by 1 and a vertical shift upwards 3 seems evident, the equation Sam graphed is:
Answer: \(y=\sqrt[3]{x-1}+3\).