To determine the equation of the cube root function that Sam graphed, we need to analyze the characteristics of the cube root function. The general form of a cube root function is:
\[ y = a \sqrt[3]{x - h} + k \]
where \((h, k)\) is the horizontal and vertical shift, respectively, and \(a\) affects the vertical stretch or compression.
Without the image, I can't directly see the plotted curve, but I can help you interpret from the options given.
Let's analyze the options provided based on their structure:
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\(y = 2\sqrt[3]{x - 1} + 3\) - This indicates that the graph has been shifted to the right by 1 unit and upwards by 3 units, with a vertical stretch factor of 2.
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\(y = \sqrt[3]{x - 1} + 3\) - This graph has the same horizontal and vertical shifts as the first option but without vertical stretching (stretched factor 1).
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\(y = \sqrt[3]{x + 1} - 3\) - Here, the graph has shifted to the left by 1 unit and downwards by 3 units.
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\(y = 2\sqrt[3]{x + 1} - 3\) - This equation keeps the shift from the previous option but adds vertical stretching by a factor of 2.
To identify the equation that best matches the plotted curve:
- If the curve generally appears centered around the point \((1, 3)\) or \((-1, -3)\) on the graph, that will help indicate the horizontal and vertical shifts.
- If the curve appears to steepen (i.e., stretch) at certain points, that would indicate the presence of the coefficient (2 in the first and last options).
If you still have the image or can provide more context or key points on the curve, I can refine the response further! Otherwise, please refer back to the visual cues in your plot to identify relevant shifts and stretch factors for the best match.