To convert a quadratic function to its vertex form, we typically start with a standard form of a quadratic equation, which is [tex]\( y = ax^2 + bx + c \)[/tex]. The vertex form of a quadratic equation is written as [tex]\( y = a(x-h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
In this particular instance, we are given that the vertex form of the function is:
[tex]\[ y = (x+\square)^2+\square \][/tex]
The vertex form already reveals certain characteristics about the function, specifically the location of its vertex. Notice the following:
1. The term inside the parentheses [tex]\((x + \square)\)[/tex] suggests a horizontal shift. The vertex occurs where the squared term is zero. Therefore, [tex]\( x = -\square \)[/tex].
2. The term outside of the parentheses, [tex]\(+ \square\)[/tex], dictates the vertical shift. This term is the y-coordinate of the vertex.
Given the problem statement, we can identify the coordinates of the vertex directly:
- The horizontal shift (x-coordinate of the vertex) is [tex]\(\square\)[/tex], so [tex]\(h = -\square\)[/tex]. This translates into the expression [tex]\(x = -(-\square)\)[/tex], which simplifies to [tex]\(x = +\square\)[/tex].
- The vertical shift (y-coordinate of the vertex) is [tex]\(\square\)[/tex].
Therefore, the vertex of the function is at the point [tex]\((- \square, \square)\)[/tex]. This new vertex form [tex]\( y = (x + \square)^2 + \square \)[/tex] correctly represents the function in terms of its horizontal and vertical shifts.
In summary, the form [tex]\( y = (x + \square)^2 + \square \)[/tex] indicates a quadratic function whose vertex is located at [tex]\((- \square, \square)\)[/tex] and captures the function's transformations concisely.
