To determine the equation of the cube root function that Sam graphed, let's analyze the general form of a cube root function:
The general form of a cube root function can be expressed as:
\[ y = a \sqrt[3]{x - h} + k \]
where:
- \( a \) affects the vertical stretch or compression,
- \( (h, k) \) is the translation of the graph.
Since you described an "s-shaped curve" on the graph suggesting that the cube root function is shifted up and possibly horizontally, the term \( (x - h) \) indicates a right or left shift while the \( + k \) indicates an upward or downward shift.
Given the options provided, without the visual representation, it can be difficult to determine exactly how it's transformed, but we can analyze the functions:
-
\( y = 2\sqrt[3]{x - 1} + 3 \)
- This would stretch the graph vertically by a factor of 2 and would shift it right 1 unit and up 3 units.
-
\( y = \sqrt[3]{x - 1} + 3 \)
- This translates right 1 unit and up 3 units with no vertical stretch or compression.
-
\( y = \sqrt[3]{x + 1} - 3 \)
- This shifts left 1 unit and down 3 units.
-
\( y = 2\sqrt[3]{x + 1} - 3 \)
- This shifts left 1 unit, down 3 units, and stretches vertically by a factor of 2.
To choose the correct equation, you'd want to match the features of the curve shown in the image. Given that it’s a cube root function starting from a point and curving, either option 1 or 2 is likely since they both have upward shifts.
If the curve starts at (1, 3) and goes upwards, the correct answer is likely:
y = \sqrt[3]{x - 1} + 3 (option 2), as it matches the typical shape of a cube root function starting to rise and shifts right by 1 and up by 3.
If you have more specific coordinates from the curve or details about the graph, please share to finalize the choice!