Question
If you subtracted 3 from the exponent in an exponential function, what would happen to the graph?
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Responses
Answers
GPT-4o mini
If you subtract 3 from the exponent in an exponential function, it effectively shifts the graph vertically downward. Here’s a more detailed explanation:
1. **Original Function**: Let’s say the original exponential function is \( f(x) = a \cdot b^x \) (where \( a \) is a positive constant and \( b > 0 \)).
2. **Modified Function**: When you subtract 3 from the exponent, the new function would be \( g(x) = a \cdot b^{(x - 3)} \), which can be rewritten as \( g(x) = a \cdot b^x \cdot b^{-3} \).
3. **Effect on the Graph**: The new graph \( g(x) \) is equivalent to the original graph \( f(x) \) being horizontally shifted to the right by 3 units. This is because the transformation \( x - 3 \) indicates a rightward shift.
4. **Overall Impact**: Consequently, while the exponential function retains its shape, its starting point and position along the X-axis are altered, leading to a horizontal rightward translation of the graph.
In summary, subtracting 3 from the exponent results in a horizontal shift of the graph to the right by 3 units.
1. **Original Function**: Let’s say the original exponential function is \( f(x) = a \cdot b^x \) (where \( a \) is a positive constant and \( b > 0 \)).
2. **Modified Function**: When you subtract 3 from the exponent, the new function would be \( g(x) = a \cdot b^{(x - 3)} \), which can be rewritten as \( g(x) = a \cdot b^x \cdot b^{-3} \).
3. **Effect on the Graph**: The new graph \( g(x) \) is equivalent to the original graph \( f(x) \) being horizontally shifted to the right by 3 units. This is because the transformation \( x - 3 \) indicates a rightward shift.
4. **Overall Impact**: Consequently, while the exponential function retains its shape, its starting point and position along the X-axis are altered, leading to a horizontal rightward translation of the graph.
In summary, subtracting 3 from the exponent results in a horizontal shift of the graph to the right by 3 units.