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The quadratic formula is a powerful tool to find the zeros, or roots, of a quadratic equation. A quadratic equation is typically represented as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The quadratic formula looks like this: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

• "\(-b\)" represents the opposite of \(b\).
• "\(\pm\)" means you do the calculation twice, once with \(+\) and once with \(-\).
• The expression under the square root, \(b^2 - 4ac\), is called the discriminant. It tells us about the nature of the roots:
• If positive, there are two real roots.
• If zero, there is one real root.
• If negative, the roots are complex.

In our example, substituting \(a = 2\), \(b = -4\), and \(c = -11\) into this formula helps us find the equation's zeros: \[ x = 1 \pm \frac{\sqrt{26}}{2} \] These are the points where our function crosses the x-axis, also known as the x-intercepts.

The vertex of a parabola is a crucial point that represents the maximum or minimum value of a quadratic function. Mathematically, for \(f(x) = ax^2 + bx + c\), the x-coordinate of the vertex can be found using the formula: \[ h = -\frac{b}{2a} \] After finding \(h\), substitute back into the quadratic equation to find \(k\), the y-coordinate: \[ k = f(h) \] In our given quadratic function, \(2x^2 - 4x - 11\), we calculated: \[ h = 1 \] Then by substituting \(h\) back, we found: \[ k = -13 \] So, the vertex is \((1, -13)\). This vertex tells us two important things:

• The minimum value of the parabola is \(-13\).
• The axis of symmetry of the parabola is the line \(x = 1\).

The vertex is vital for graphing and understanding the behavior of the quadratic function.

The zeros of a function, also known as roots or solutions, are the x-values where the function equals zero. For a quadratic function, finding these zeros helps us understand where the parabola crosses the x-axis. The zeros can be found using the quadratic formula, factoring (if possible), or completing the square. For example, for the function \(f(x) = 2x^2 - 4x - 11\), the quadratic formula reveals two roots: \[ x = 1 \pm \frac{\sqrt{26}}{2} \]

• At \(x = 1 + \frac{\sqrt{26}}{2}\), the function crosses the x-axis at a positive point.
• At \(x = 1 - \frac{\sqrt{26}}{2}\), the intersection is on the negative side.

These intersections on the graph provide insight into the function's behavior across its domain. Knowing the zeros is essential for graphing the function accurately and understanding its solution set.

Graphing quadratic equations involves plotting a parabola, a u-shaped curve. The basic form of a quadratic equation is \(ax^2 + bx + c\). The graph of this equation can be plotted by identifying a few critical points:

• The vertex: Provides the highest or lowest point of the parabola.
• The zeros (or x-intercepts): Indicate where the parabola crosses the x-axis.
• The y-intercept: Found by setting \(x = 0\), and calculating \(f(0) = c\).

In the quadratic function \(f(x) = 2x^2 - 4x - 11\), the vertex is at \((1, -13)\), indicating the graph's minimum point. Additionally, since the leading coefficient \(a = 2\) is positive, the parabola opens upward. It crosses the x-axis at its calculated zeros, around the values derived from \(x = 1 \pm \frac{\sqrt{26}}{2}\). The y-intercept, where the graph crosses the y-axis, is at \(f(0) = -11\).Plotting these key points on the graph will demonstrate the parabola's "U" shape and help understand the behavior of the quadratic function.