To understand how the graph of \( g(x) = \cos(x) - 12 \) compares with the parent function \( g(x) = \cos(x) \), we can analyze the transformations that have been applied to the parent function.
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Parent Function: The parent function \( g(x) = \cos(x) \) has a range of \([-1, 1]\) and oscillates between these values with a period of \(2\pi\).
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Transformation: The function \( g(x) = \cos(x) - 12 \) involves a vertical shift:
- The term \(-12\) indicates that the entire graph of \( \cos(x) \) is shifted downward by 12 units.
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New Range: After shifting down by 12, the new range of \( g(x) = \cos(x) - 12 \) becomes:
- The maximum value shifts from 1 to \(1 - 12 = -11\).
- The minimum value shifts from -1 to \(-1 - 12 = -13\).
- Therefore, the new range is \([-13, -11]\).
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Period: The period of the cosine function is unchanged, so the period of \( g(x) = \cos(x) - 12 \) remains \(2\pi\).
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Graph Comparison:
- When comparing the two graphs, the graph of \( g(x) = \cos(x) \) oscillates between -1 and 1 centered around the x-axis.
- The graph of \( g(x) = \cos(x) - 12 \) will oscillate between -13 and -11, centered around the line \( y = -12 \).
Conclusion: The graph of \( g(x) = \cos(x) - 12 \) is the same shape as the graph of the parent function \( g(x) = \cos(x) \), with both graphs oscillating with a period of \(2\pi\), but the entire graph is moved down vertically by 12 units.
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