To find the values of [tex]\(a\)[/tex], [tex]\(p\)[/tex], and [tex]\(q\)[/tex] for the transformed parabola given the initial form [tex]\(y = x^2\)[/tex] and the specified transformations, let's analyze each transformation step by step:
1. Vertical Expansion by a Factor of 4:
- The factor that determines the vertical scaling is denoted by [tex]\(a\)[/tex].
- Since the parabola is expanded vertically by a factor of 4, the value of [tex]\(a\)[/tex] must be 4.
2. Horizontal Translation 3 Units Left:
- Horizontal translation shifts the parabola left or right.
- The general form [tex]\(y = a(x - p)^2 + q\)[/tex] takes [tex]\(p\)[/tex] as the horizontal shift.
- Moving the parabola 3 units to the left means [tex]\(p\)[/tex] is -3 (since leftward shifts are represented by a negative number).
3. Vertical Translation 2 Units Down:
- Vertical translation shifts the parabola up or down.
- In the form [tex]\(y = a(x - p)^2 + q\)[/tex], [tex]\(q\)[/tex] represents the vertical translation.
- Moving the parabola 2 units down means [tex]\(q\)[/tex] is -2 (since downward shifts are represented by a negative number).
Based on these transformations, we obtain the following values:
- [tex]\(a = 4\)[/tex]
- [tex]\(p = -3\)[/tex]
- [tex]\(q = -2\)[/tex]
Thus, the correct values are:
[tex]$ a = 4, \, p = -3, \, q = -2 $[/tex]
Therefore, the correct choice is:
[tex]\[ \boxed{a=4, p=-3, q=-2} \][/tex]
