Problem 96
Question
Begin by graphing the standard cubic function, \(f(x)=x^{3} .\) Then use transformations of this graph to graph the given function. $$g(x)=x^{3}-2$$
Step-by-Step Solution
Verified Answer
The graph of the function \(g(x) = x^3 - 2\) will have the same shape as the standard cubic function \(f(x) = x^3\), but it will be shifted down by 2 units due to the transformation indicated by -2.
1Step 1: Graph the Standard Cubic Function
Start off by sketching the graph of the standard cubic function \(f(x) = x^3\). This is a simple cubic curve that passes through the origin and increases to the right and left of the Y-axis.
2Step 2: Identify the Transformation
Notice in the function \(g(x) = x^3 - 2\), compared to the standard cubic function, a -2 is subtracted. This is a vertical shift or translation. In this case, it means that every point on the original graph of \(f(x) = x^3\) will be shifted downwards by 2 units to generate the graph for \(g(x)\).
3Step 3: Graph the Transformed Function
Finally, sketch the graph for function \(g(x) = x^3 - 2\). Begin with the graph for \(f(x) = x^3\) and shift every point on the graph downwards by 2 units. Make sure to indicate this shift on the graph.
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Problem 96
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