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A quadratic function is expressed in the form \[f(x) = ax^2 + bx + c\], where

• \(a\), \(b\), and \(c\) are coefficients
• \(a\) cannot be zero,

The equation \(f(x) = x^2 - 8x + 8\) fits into this format seamlessly, thus representing the standard form of a quadratic function.
Being in this form makes it easier to recognize the various components of the function, which are crucial for understanding its behavior and shape:

• The coefficient \(a = 1\) determines how "wide" or "narrow" the parabola is.
• The coefficient \(b = -8\) influences the symmetry of the parabola.
• The constant \(c = 8\) gives the y-intercept of the graph.

Recognizing these elements helps predict the general shape of the parabola and supports further transformation to find other forms or characteristics of the quadratic function.

The vertex form of a quadratic function is given by \[f(x)= a(x-h)^2 + k\], where

• \((h, k)\) is the vertex of the parabola,
• \(h\) represents the x-coordinate of the vertex,
• \(k\) represents the y-coordinate, giving the maximum or minimum value.

For the function \(f(x) = x^2 - 8x + 8\), you can determine the vertex using the formula \(x = -\frac{b}{2a}\), where the solution for \(x\) gives us the x-coordinate of the vertex.
By substituting the identified values of \(a = 1\) and \(b = -8\), we calculate:\[x = -\frac{-8}{2 \times 1} = 4\]Plugging \(x = 4\) back into the original function develops the complete vertex coordinate:\[k = f(4) = (4)^2 - 8\times4 + 8 = -8\]This reveals that the vertex is at \((4, -8)\), implying that the function can be rewritten as \[f(x) = (x - 4)^2 - 8\].
This form is particularly useful as it clearly highlights the turning point of the quadratic function.

Quadratic functions reach a maximum or minimum value at their vertex. The coefficient \(a\) in the standard form determines the direction of the parabola:

• If \(a > 0\), the parabola opens upwards, indicating that the vertex is a minimum point.
• If \(a < 0\), the parabola opens downwards, so the vertex represents a maximum.

For our function \(f(x) = x^2 - 8x + 8\), with \(a = 1\) being positive, the parabola opens upwards, and consequently, the vertex \((4, -8)\) signifies the minimum point.
This insight helps not only in pinpointing the vertex but also in understanding the behavior of the entire graph.The minimum value of the function is \(-8\), which occurs when \(x = 4\).
This property is particularly helpful when tackling optimization problems, as it allows you to readily identify extreme values courtesy of the vertex.