Discover answers to your questions with Westonci.ca, the leading Q&A platform that connects you with knowledgeable experts. Explore in-depth answers to your questions from a knowledgeable community of experts across different fields. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.

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

A. [tex]a=4, p=2, q=3[/tex]
B. [tex]a=4, p=-2, q=-3[/tex]
C. [tex]a=4, p=-3, q=-2[/tex]
D. [tex]a=-3, p=2, q=4[/tex]

Sagot :

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]
We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.