91Ó°ÊÓ

The vertex of a parabola is a crucial point. It represents the maximum or minimum value of the quadratic function. For functions in the form of \(f(x) = ax^2 + bx + c\), we can find the x-coordinate of the vertex using the formula \(x = -\frac{b}{2a}\). This formula is derived from the process of completing the square. By substituting the values of \(a\) and \(b\), we can easily calculate the x-coordinate of the vertex.

For the function \(f(x) = 2x^2 + 12x - 3\), we identified that \(a = 2\) and \(b = 12\). Plugging these into the formula gives \(x = -\frac{12}{2(2)} = -3\). Hence, the vertex occurs at \(x = -3\).

But we are not done yet; we also need the y-coordinate, which we get by substituting \(x = -3\) back into the original function. By calculating \(f(-3)\), we find that the y-coordinate of the vertex is \(-21\). Therefore, the vertex of the parabola is \((-3, -21)\).

Quadratic functions can have either a minimum or maximum value, depending on their orientation. The orientation is determined by the coefficient of the \(x^2\) term, denoted as \(a\).

If \(a > 0\), the parabola opens upwards, resembling a 'U' shape, indicating a minimum value at the vertex. Conversely, if \(a < 0\), the parabola opens downwards, resembling an upside-down 'U', indicating a maximum value at the vertex.

In our example, \(f(x) = 2x^2 + 12x - 3\), since \(a = 2\) which is positive, the parabola opens upwards. This means the function has a minimum value at the vertex. We previously determined the vertex to be at \((x,y) = (-3, -21)\). Therefore, the minimum value of the function is \(-21\). Remember, the y-coordinate of the vertex always represents the minimum or maximum value of the function.

Solving quadratic equations involves finding the values of \(x\) that make the function \(f(x) = 0\). There are various methods to solve quadratic equations: factoring, using the quadratic formula, completing the square, or graphing.

The Quadratic Formula is especially useful as it works for any quadratic equation. The formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula gives the roots or solutions of the quadratic equation \(ax^2 + bx + c = 0\).

Let's look at our function \(f(x) = 2x^2 + 12x - 3\). To find the roots, we would set it equal to zero: \(2x^2 + 12x - 3 = 0\). Use the quadratic formula where \(a = 2\), \(b = 12\), and \(c = -3\). Plugging in these values will give the solutions for \(x\).

It is crucial to know that the roots of the equation where \(f(x) = 0\) tell us the x-intercepts of the parabola. These points indicate where the function crosses the x-axis. Combining this with understanding the vertex helps in sketching the graph of the quadratic function fully.