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Describe how to move the graph of m(x) so that it coincides with the graph of p(x).

The graphs of \( m(x) \) and \( p(x) \) shown in the image are parabolas. To align or make the graph of \( m(x) \) coincide with \( p(x) \), you would need to apply the following transformations:

1. **Vertical Translation**: 
   - The vertex of \( p(x) \) is higher than the vertex of \( m(x) \). You need to shift \( m(x) \) upwards until their vertices align. The exact value of this vertical shift is the difference between the y-coordinates of the vertices of \( p(x) \) and \( m(x) \).

2. **Horizontal Translation**:
   - The graphs appear to have their vertices along the same vertical line, which means no horizontal translation is necessary.

Thus, the transformation required is a **vertical translation upward** by a certain amount, which is determined by the vertical distance between the vertex of \( m(x) \) and \( p(x) \).

Would you like to know more details or have any questions?

Here are five related questions to expand on this:

1. How can we determine the exact value of the vertical translation needed?
2. What is the general form of a vertical translation for any function?
3. How does a horizontal translation differ from a vertical one in terms of the function's equation?
4. What are the effects of combining vertical and horizontal translations on a graph?
5. How can you verify that the transformation was applied correctly?

**Tip:** When shifting graphs, always compare the coordinates of key points like vertices to determine the necessary translations.

Discover how to align the graph of m(x) with p(x) using vertical translation. Learn the key steps and concepts involved in shifting one parabola to coincide with another. Suitable for high school students studying graph transformations.

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