91Ó°ÊÓ

The zeros of a function, often called the roots or solutions, are the values of \(x\) for which the function \(f(x)\) equals zero. For a quadratic function, finding these zeros is essential because it tells us the points where the graph intersects the x-axis.
The function we are dealing with is \(f(x) = 2x^2 - 4x - 11\). This is a quadratic function, so we use the quadratic formula to find the zeros:

• The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Identify the coefficients: \(a = 2\), \(b = -4\), \(c = -11\).
• Plug these into the formula: \(x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \times 2 \times (-11)}}{2 \times 2}\).

Calculate the discriminant, \(b^2 - 4ac = 104\). The discriminant being positive implies two real and distinct zeros.
After simplifying, we find the zeros: \(x = 1 + \frac{\sqrt{26}}{2}\) and \(x = 1 - \frac{\sqrt{26}}{2}\). These are the points where the parabola crosses the x-axis.

The vertex of a parabola is a pivotal point because it represents the maximum or minimum value of a quadratic function. It serves as the turning point of the parabola.
To find the vertex, we use the formula for the x-coordinate: \(h = -\frac{b}{2a}\).
For \(f(x) = 2x^2 - 4x - 11\), the calculation goes as follows:

• Calculate \(h = \frac{4}{4} = 1\).

Now, substitute \(x = 1\) back into the function to find the y-coordinate:
\(f(1) = 2(1)^2 - 4(1) - 11 = -13\).
Thus, the vertex is at \((1, -13)\). Since the coefficient of \(x^2\) (i.e., \(a = 2\)) is positive, this point is a minimum. The parabola opens upwards, making this minimum the lowest point on the graph.

Graphing a quadratic function helps visualize the function's behavior, showing the symmetry and direction in which the parabola opens.
For \(f(x) = 2x^2 - 4x - 11\), follow these steps to sketch the graph:

• The parabola opens upwards because \(a = 2\) is positive.
• Vertex: We've found the vertex to be \((1, -13)\).
• Intercepts:
  • X-intercepts are the zeros of the function: \(x = 1 + \frac{\sqrt{26}}{2}\) and \(x = 1 - \frac{\sqrt{26}}{2}\).
  • Find the y-intercept by setting \(x = 0\): \(f(0) = -11\).

To graph, plot the vertex, the x-intercepts, and the y-intercept. Draw the parabola symmetrically about the vertex to complete the graph. Ensure the parabola extends upwards, confirming its shape corresponds to the given quadratic function.