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Write the function for p(x) in terms of m(x).

To express \( p(x) \) in terms of \( m(x) \), we need to identify the transformation that maps \( m(x) \) to \( p(x) \).

From the image, we can observe that:

1. **Vertical Translation**: The graph of \( p(x) \) is a vertically shifted version of \( m(x) \). The vertex of \( p(x) \) is above the vertex of \( m(x) \). To determine the exact translation, we count the vertical distance between the two vertices.

   It appears that the vertex of \( p(x) \) is 5 units above the vertex of \( m(x) \).

Given this observation, the function \( p(x) \) can be written as:
\[
p(x) = m(x) + 5
\]

This equation indicates that the function \( p(x) \) is obtained by adding 5 to \( m(x) \), which corresponds to a vertical shift of 5 units upward.

Would you like further details or have any questions?

Here are five related questions to expand on this:

1. How would the function change if \( p(x) \) was shifted downward instead of upward?
2. How can you verify that \( p(x) = m(x) + 5 \) by substituting specific values of \( x \)?
3. What other types of transformations can be applied to \( m(x) \) to obtain different functions?
4. How would the equation change if the shift was horizontal instead of vertical?
5. Can you derive the equation for \( m(x) \) in terms of \( p(x) \) if you know \( p(x) = m(x) + 5 \)?

**Tip:** Always compare the positions of corresponding points (like vertices) on the graphs when determining how to express one function in terms of another.

Understand how to express the function p(x) in terms of m(x) with a vertical shift explanation. Learn the step-by-step process of identifying and applying vertical translations in function transformations. Suitable for high school students studying functions and graph transformations.

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