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Let's start with understanding the vertex formula. The vertex of a quadratic function is the highest or lowest point on its graph, known as a parabola. This can be useful for finding the maximum or minimum value of the function. Given a quadratic function in the general form \(y = ax^2 + bx + c\), the x-coordinate of the vertex can be found using the vertex formula: \[-\frac{b}{2a}\]. Here, \(a\) and \(b\) are coefficients of the quadratic and linear terms respectively. Once you have the x-coordinate, you can find the y-coordinate by substituting this x value back into the original equation. For instance, if \(y = x^2 + 8x - 3,\) the vertex formula gives us \(x = -\frac{8}{2} = -4\). Plugging \(x = -4\) back into the equation gives us the y-coordinate. Hence, the vertex is \((-4, -19)\).

A quadratic function is any function that can be written in the form \(y = ax^2 + bx + c\), where \(a, b,\) and \(c\) are constants, and \(a eq 0\). The graph of a quadratic function is a parabola, which can either open upwards (if \(a > 0\)) or downwards (if \(a < 0\)). The parabola's shape is determined by the value of \(a\), its position can be shifted based on values of \(b\) and \(c\). Key features of a quadratic function's graph include:

• Vertex, which is the highest or lowest point
• Axis of symmetry, a vertical line passing through the vertex
• Y-intercept, where the graph crosses the y-axis
• Finding these features help in graphing or understanding the quadratic function better.

The coefficients \(a, b,\) and \(c\) in a quadratic equation \(y = ax^2 + bx + c\) have specific roles:

• Coefficient \(a\): Determines the parabola's direction and width. If \(|a| > 1,\) the parabola is narrower. If \(|a| < 1,\) it is wider. Positive \(a\) values make the parabola open upwards, while negative \(a\) values make it open downwards.
• Coefficient \(b\): Influences the position of the axis of symmetry and the vertex. It affects where the peak or trough of the parabola occurs horizontally.
• Coefficient \(c\): Represents the y-intercept, where the parabola crosses the y-axis. The value of \(c\) gives the function's starting point when \(x = 0\).

In the given exercise \(y = x^2 + 8x - 3\), \(a = 1\), \(b = 8\), and \(c = -3\). Identifying these coefficients helps in easily finding the vertex, axis of symmetry, and other characteristics of the quadratic function.

Completing the square is another method to find the vertex of a quadratic function. It involves rewriting the quadratic function in vertex form \(y = a(x-h)^2 + k\), where \(h\) and \(k\) are the coordinates of the vertex. Here’s how you do it:

• Start with the standard form \(y = ax^2 + bx + c\).
• Factor out \(a\) from the first two terms: \(y = a(x^2 + \frac{b}{a}x) + c\).
• Add and subtract \(\left( \frac{b}{2a} \right)^2\) inside the parenthesis: \(y = a(x^2 + \frac{b}{a}x + \left( \frac{b}{2a} \right)^2 - \left( \frac{b}{2a} \right)^2) + c\).
• Rewrite the perfect square trinomial: \(y = a\left( x + \frac{b}{2a} \right)^2 - \frac{b^2}{4a} + c\).
• Simplify to identify \(h = -\frac{b}{2a}\) and \(k = c - \frac{b^2}{4a}\).

This method can be seen as an alternate to applying the vertex formula directly. The answer will be the same, giving the vertex as \((-4, -19)\). This approach ensures a deeper understanding of how quadratic functions and their graphs work.