Question
How does the graph of g(x) = (x + 2)3 − 7 compare to the parent function of f(x) = x3?
g(x) is shifted 2 units to the right and 7 units down.
g(x) is shifted 7 units to the right and 2 units up.
g(x) is shifted 2 units to the left and 7 units down.
g(x) is shifted 7 units to the left and 2 units down.
g(x) is shifted 2 units to the right and 7 units down.
g(x) is shifted 7 units to the right and 2 units up.
g(x) is shifted 2 units to the left and 7 units down.
g(x) is shifted 7 units to the left and 2 units down.
Answers
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Answered by
GPT-4o mini
To analyze how the graph of \( g(x) = (x + 2)^3 - 7 \) compares to the parent function \( f(x) = x^3 \), we can identify the transformations involved.
1. **Shift to the left/right**: The expression \( (x + 2)^3 \) indicates a horizontal shift. In general, if we have \( (x - h)^3 \), the graph shifts \( h \) units to the right if \( h \) is positive and \( |h| \) units to the left if \( h \) is negative. In this case, \( (x + 2) \) means the graph is shifted 2 units to the left (since \( h = -2 \)).
2. **Shift up/down**: The \( -7 \) at the end of the function indicates a vertical shift. If we have \( f(x) - k \), where \( k \) is positive, the graph shifts \( k \) units down. Here, \( g(x) = (x + 2)^3 - 7 \) means the graph is shifted 7 units down.
Combining these two observations, we conclude that:
- The graph of \( g(x) \) is shifted **2 units to the left** and **7 units down** compared to the parent function \( f(x) = x^3 \).
Thus, the correct answer is:
**g(x) is shifted 2 units to the left and 7 units down.**
1. **Shift to the left/right**: The expression \( (x + 2)^3 \) indicates a horizontal shift. In general, if we have \( (x - h)^3 \), the graph shifts \( h \) units to the right if \( h \) is positive and \( |h| \) units to the left if \( h \) is negative. In this case, \( (x + 2) \) means the graph is shifted 2 units to the left (since \( h = -2 \)).
2. **Shift up/down**: The \( -7 \) at the end of the function indicates a vertical shift. If we have \( f(x) - k \), where \( k \) is positive, the graph shifts \( k \) units down. Here, \( g(x) = (x + 2)^3 - 7 \) means the graph is shifted 7 units down.
Combining these two observations, we conclude that:
- The graph of \( g(x) \) is shifted **2 units to the left** and **7 units down** compared to the parent function \( f(x) = x^3 \).
Thus, the correct answer is:
**g(x) is shifted 2 units to the left and 7 units down.**
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