The parabola y=x^2 is changed to the form y=a(x-p)^2 +q by translating the parabola 2 units up and 4 units right and expanding it vertically by a factor of 3 what are the values of a,p and q

The general form of the parabola is transformed from \( y = x^2 \) to \( y = a(x - p)^2 + q \) by applying a vertical expansion, as well as horizontal and vertical translations.

### Steps to find \( a \), \( p \), and \( q \):

1. **Translation 4 units to the right**: 
   When a parabola is translated horizontally to the right by \( 4 \) units, the \( x \)-term becomes \( (x - 4) \), so the equation becomes:
   \[
   y = a(x - 4)^2 + q
   \]

2. **Translation 2 units up**: 
   Translating vertically 2 units up increases the \( y \)-intercept by 2. So the equation now is:
   \[
   y = a(x - 4)^2 + 2
   \]

3. **Vertical expansion by a factor of 3**: 
   Expanding vertically by a factor of 3 means multiplying the entire function by 3. Since the original parabola \( y = x^2 \) has \( a = 1 \), after the expansion, \( a = 3 \). So the equation now becomes:
   \[
   y = 3(x - 4)^2 + 2
   \]

Thus, the transformed equation is \( y = 3(x - 4)^2 + 2 \).

### Final values of \( a \), \( p \), and \( q \):
- \( a = 3 \)
- \( p = 4 \)
- \( q = 2 \)

Would you like a detailed breakdown of any of these steps?

Here are 5 related questions:
1. How would the equation change if the parabola were translated 3 units left instead of right?
2. What happens to the vertex of the parabola under these transformations?
3. How does the graph of \( y = 3(x - 4)^2 + 2 \) compare to the graph of \( y = x^2 \)?
4. How does a negative value of \( a \) affect the parabola's shape?
5. What if the vertical expansion factor was 1/2 instead of 3?

**Tip:** The parameter \( a \) controls the "width" and direction of the parabola—larger values of \( a \) make the parabola narrower, while smaller values make it wider.

The parabola y=x^2 is changed to the form y=a(x-p)^2 +q by translating the parabola 2 units up and 4 units right and expanding it vertically by a factor of 3. What are the values of a, p, and q?

Learn how to transform the parabola y = x^2 by translating it 2 units up, 4 units right, and expanding it vertically by a factor of 3. This problem demonstrates key quadratic concepts, including translation, vertical stretch, and finding the parameters a, p, and q. Suitable for Grades 9-11.

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