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The quadratic formula is a powerful tool to find the zeros, or roots, of a quadratic function. It is given by the formula - \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) This formula helps us solve any quadratic equation of the form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are coefficients from the quadratic expression.
For the function \(f(x) = 2x^2 - 4x - 11\): - We identify the coefficients as: \(a = 2\), \(b = -4\), and \(c = -11\). Substituting these values into the quadratic formula allows us to find the zeros.
The discriminant \(b^2 - 4ac\) determines the nature of the roots:

• If it is positive, the equation has two distinct real roots.
• If zero, there is exactly one real root.
• If negative, the roots are complex.

In this exercise, solving gives two real and distinct roots: \(x = 1 \pm \frac{\sqrt{26}}{2}\).

The vertex of a parabola is an important point as it defines the extremum (maximum or minimum) of the function. To find the vertex of a quadratic function represented as \(ax^2 + bx + c\), use the formula for the x-coordinate: - \(x = \frac{-b}{2a}\)
In the given function \(f(x) = 2x^2 - 4x - 11\), plugging \(b = -4\) and \(a = 2\) yields the x-coordinate of the vertex: - \(x = \frac{4}{4} = 1\) Next, substitute \(x = 1\) back into the function to find the y-coordinate of the vertex, \(f(1)\): - \(f(1) = 2 \times 1^2 - 4 \times 1 - 11 = -13\)
Thus, the vertex is at \((1, -13)\). In this case, because the quadratic coefficient \(a = 2\) is positive, the parabola opens upwards, indicating that the vertex is a minimum point.

The zeros of a quadratic function, also referred to as the roots or solutions, are the x-values where the function equals zero. These points can also be seen as where the parabola intersects the x-axis. In a standard quadratic equation \(ax^2 + bx + c = 0\), the zeros can be found using the quadratic formula.
For instance, with \(f(x) = 2x^2 - 4x - 11\), the quadratic formula provides us the solutions or zeros:

• \(x = 1 + \frac{\sqrt{26}}{2}\)
• \(x = 1 - \frac{\sqrt{26}}{2}\)

These zeros tell us the points on the graph where it will cross the x-axis, giving visual insight into where the function impacts zero. Recognizing these points is crucial when graphing the function or solving quadratic equations graphically.

Graphing a parabola involves visualizing the quadratic function on a coordinate plane. To effectively sketch the graph of \(f(x) = 2x^2 - 4x - 11\), a few key points and properties must be considered:
1. **Vertex**: The point \((1, -13)\) represents the lowest point on the graph since the parabola opens upwards. 2. **Zeros**: These are approximately \(x = 1 + \frac{\sqrt{26}}{2}\) and \(x = 1 - \frac{\sqrt{26}}{2}\), showing where the graph cuts through the x-axis. 3. **Direction**: Since \(a = 2\) is positive, the parabola opens upwards, resembling a U shape.
To sketch, plot the vertex and zeros on the graph. Use symmetry around the vertex line to guide the shape. Remember, a parabola is a symmetric curve, so ensure both sides mirror each other. Drawing a smooth curve through the vertex and zeros completes the sketch. This creates a complete picture of how the quadratic function behaves.