To solve the problem and identify the vertex and transformations for the given equation [tex]\( y = \frac{1}{3} x^2 \)[/tex], let's follow these steps:
1. Vertex Identification:
Any quadratic equation of the form [tex]\( y = a(x-h)^2 + k \)[/tex] has its vertex at the point [tex]\((h, k)\)[/tex]. This is known as the vertex form of a quadratic equation.
The given equation is [tex]\( y = \frac{1}{3} x^2 \)[/tex].
- Here, [tex]\( a = \frac{1}{3} \)[/tex], [tex]\( h = 0 \)[/tex], and [tex]\( k = 0 \)[/tex]. Therefore, the equation is in the form [tex]\( y = a(x - 0)^2 + 0 \)[/tex].
- This simplifies to [tex]\( y = \frac{1}{3} x^2 \)[/tex].
The vertex of this equation is [tex]\((h, k) = (0, 0)\)[/tex].
2. Transformation Description:
The coefficient [tex]\(\frac{1}{3}\)[/tex] in front of [tex]\( x^2 \)[/tex] describes a vertical transformation.
- When this coefficient is between 0 and 1 (i.e., [tex]\(0 < a < 1\)[/tex]), it indicates a vertical compression.
- Conversely, if the coefficient were greater than 1, it would indicate a vertical stretch.
In this case, [tex]\( a = \frac{1}{3} \)[/tex], which is less than 1 but greater than 0, so it represents a vertical compression.
To summarize, the vertex of the equation [tex]\( y = \frac{1}{3} x^2 \)[/tex] is [tex]\((0, 0)\)[/tex] and the transformation is a vertical compression.
The correct answer is:
- [tex]\((0,0)\)[/tex]; vertical compression
So, the correct choice among the given options is:
a [tex]\((0,0)\)[/tex]: vertical compression
