91Ó°ÊÓ

The vertex form of a quadratic function makes it easy to identify the vertex of the parabola, which is either a maximum or a minimum point depending on the direction the parabola opens. In general, the vertex form is given by:\[ g(x) = a(x-h)^2 + k \]Here:

• \(a\) determines the direction and the width of the parabola.
• \(h\) and \(k\) are the coordinates of the vertex \((h, k)\).

When converting a quadratic from its standard form \(ax^2 + bx + c\) to the vertex form, completing the square is a common method used. The vertex form is particularly useful because it gives you at a glance:

• The vertex of the quadratic function.
• The axis of symmetry of the parabola \(x=h\), which cuts the parabola into two symmetric halves.

In the context of the problem, although we expanded to find the standard form initially, recognizing the vertex form directly allows for quick determination of the vertex.

The minimum value of a quadratic function is the smallest value that the function can take. For quadratic functions of the form \(g(x) = ax^2 + bx + c\), the minimum or maximum value occurs at the vertex. To find the vertex, use the formula for the \(x\)-coordinate:\[ x = -\frac{b}{2a} \]With \(a = 2\) and \(b = -8\) from the problem, this leads to \(x = 2\). Given that \(a > 0\) (which is true for this problem), the parabola opens upwards. This means the vertex is the point where the minimum value occurs. Once you have the \(x\)-coordinate of the vertex, substitute it back into the original equation to find the corresponding \(y\)-coordinate or minimum value:\[ g(2) = 2(2)^2 - 8(2) + 7 \]Simplifying this gives a minimum value of \(g(2) = -1\). This vertex represents the lowest point on the curve.

Expanding a quadratic expression involves removing parentheses and simplifying the expression to the standard form \(ax^2 + bx + c\). For the problem at hand, the given expression was \(g(x) = 2x(x-4) + 7\). To expand it, distribute the \(2x\) across the \((x-4)\):

• \(2x \times x = 2x^2\)
• \(2x \times -4 = -8x\)

Thus, the expanded form becomes:\[ g(x) = 2x^2 - 8x + 7 \]Expanding is a crucial step because it presents the quadratic expression in a form that is more straightforward for analyzing properties such as the vertex and intercepts. When working with quadratics, expansion helps in spotting the coefficients \(a\), \(b\), and \(c\) directly, which are essential for finding the vertex and determining the function's maximum or minimum value. Remember, expansion transforms complex expressions into a simpler form, enabling ease of further algebraic manipulations.