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The vertex of a parabola is a crucial point, often referred to as the "turning point" because it is where the graph changes direction. For a parabola, which is the graph of a quadratic function, the vertex can either be the minimum or maximum point. This depends on whether the parabola opens upwards or downwards:

• If the parabola opens upwards, the vertex is the lowest point, representing the minimum value.
• If it opens downwards, the vertex is the highest point, indicating the maximum value.

For the quadratic function we examined, given by \( f(x) = x^2 - 6x + 2 \), we find the vertex using the formula \( h = \frac{-b}{2a} \). Here, the coefficients are identified as \( a = 1 \), \( b = -6 \), and \( c = 2 \). Substituting these into the formula gives the x-coordinate of the vertex as \( h = 3 \).
Once we have the x-coordinate, we substitute it back into the original quadratic function to find the y-coordinate. Thus, \( k = f(3) = -7 \). Therefore, the vertex is at \((3, -7)\), and since \( a = 1 \) (positive), the parabola opens upwards, confirming the vertex is indeed a point of minimum value.

The minimum value of a quadratic function is an important feature and relates directly to its vertex. In standard form, a quadratic function is written as \( f(x) = ax^2 + bx + c \), and when \( a > 0 \), the function has a minimum value at its vertex. This is because the parabola opens upwards,
implying that all other points on the parabola are at a higher y-value than the vertex.To find the minimum value, we evaluate the function at the vertex. In our specific example, we already calculated the vertex to be at \( (3, -7) \), indicating the minimum value of the function \( f(x) = x^2 - 6x + 2 \) is \(-7\).

• This minimum value corresponds to the y-coordinate of the vertex.
• Since the parabola represents every possible value of the function's expressions in this range, the vertex gives us this lowest possible value.

Thus, the minimum value can be found simply by identifying the vertex and evaluating the function at that point.

Rewriting a quadratic equation in vertex form offers clear insights into the function's graph, particularly its vertex. The vertex form is expressed as \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola.To convert a standard form quadratic \( f(x) = ax^2 + bx + c \) into vertex form, we find the vertex using \( h = \frac{-b}{2a} \) and \( k = f(h) \). With our quadratic \( f(x) = x^2 - 6x + 2 \), this transformation revealed the vertex to be at \( (3, -7) \), placing the vertex form equation as:

• \( f(x) = 1(x - 3)^2 - 7 \)

This expression highlights the vertex directly, allowing us to quickly infer the minimum point and the direction of the parabola.
By manipulating the original equation into vertex form, we gain easier access to understanding the nature of the quadratic graph, leading to quicker insights into the behavior of the function.