Question
there is a graph with points at (-1,2) (0,4) and (1,8). this is a dilation of the original function y=2^x. explain the transformation
Answers
GPT-4o mini
To understand the transformation that describes the dilation of the function \( y = 2^x \) to the points \((-1, 2)\), \((0, 4)\), and \((1, 8)\), we can follow these steps:
1. **Identify the original function**: The original function is \( y = 2^x \). We can evaluate this function at several points:
- \( y(0) = 2^0 = 1 \)
- \( y(1) = 2^1 = 2 \)
- \( y(-1) = 2^{-1} = \frac{1}{2} \)
So, the original function \( y = 2^x \) passes through the points:
- \((-1, 0.5)\)
- \((0, 1)\)
- \((1, 2)\)
2. **Examine the new points**: The new points are \((-1, 2)\), \((0, 4)\), and \((1, 8)\).
3. **Identify the transformation**: To relate these new points to the original function, we can observe the following:
- Point \((-1, 2)\): This can be seen as \( \left(-1, 4 \cdot \frac{1}{2} \right) \).
- Point \((0, 4)\): This represents \( 4 \cdot 1 \).
- Point \((1, 8)\): This can be seen as \( 4 \cdot 2 \).
4. **General transformation**: The new function can be expressed as \( y = 4 \cdot 2^x \). This indicates that every output of the original function \( y = 2^x \) is multiplied by 4.
5. **Conclusion**: The transformation from \( y = 2^x \) to the new points is a vertical dilation (stretch) by a factor of 4. The points on the graph of \( y = 2^x \) have been stretched vertically to reach the new coordinates.
In summary, the transformation involved in moving from the function \( y = 2^x \) to the points \((-1, 2)\), \((0, 4)\), and \((1, 8)\) is a vertical dilation by a factor of 4, resulting in the new function \( y = 4 \cdot 2^x \).
1. **Identify the original function**: The original function is \( y = 2^x \). We can evaluate this function at several points:
- \( y(0) = 2^0 = 1 \)
- \( y(1) = 2^1 = 2 \)
- \( y(-1) = 2^{-1} = \frac{1}{2} \)
So, the original function \( y = 2^x \) passes through the points:
- \((-1, 0.5)\)
- \((0, 1)\)
- \((1, 2)\)
2. **Examine the new points**: The new points are \((-1, 2)\), \((0, 4)\), and \((1, 8)\).
3. **Identify the transformation**: To relate these new points to the original function, we can observe the following:
- Point \((-1, 2)\): This can be seen as \( \left(-1, 4 \cdot \frac{1}{2} \right) \).
- Point \((0, 4)\): This represents \( 4 \cdot 1 \).
- Point \((1, 8)\): This can be seen as \( 4 \cdot 2 \).
4. **General transformation**: The new function can be expressed as \( y = 4 \cdot 2^x \). This indicates that every output of the original function \( y = 2^x \) is multiplied by 4.
5. **Conclusion**: The transformation from \( y = 2^x \) to the new points is a vertical dilation (stretch) by a factor of 4. The points on the graph of \( y = 2^x \) have been stretched vertically to reach the new coordinates.
In summary, the transformation involved in moving from the function \( y = 2^x \) to the points \((-1, 2)\), \((0, 4)\), and \((1, 8)\) is a vertical dilation by a factor of 4, resulting in the new function \( y = 4 \cdot 2^x \).