91Ó°ÊÓ

In the context of quadratic functions, the vertex is an important point on the graph of a parabola. It represents either the highest or lowest point, depending on whether the parabola opens upward or downward. For a function given in the form of \( ax^2 + bx + c \), the formula to find the x-coordinate of the vertex is \( x = -\frac{b}{2a} \).
Once the x-coordinate is known, substitute it back into the function to find the y-coordinate. This will give the complete vertex point \((x, y)\).
In our problem, we used the function \( f(x) = 2x^2 + 4x - 1 \), with coefficients \( a = 2 \), \( b = 4 \), and \( c = -1 \).
This led to the vertex calculation \( x = -\frac{4}{4} = -1 \). Then, by substituting \( x = -1 \), we got the vertex at \((-1, -3)\).
Remembering the vertex is crucial in understanding the basic shape and position of the parabola on a graph.

Finding the x-intercepts of a quadratic function involves determining the points where the graph crosses the x-axis. These are the solutions to the equation \( f(x) = 0 \).
For a quadratic equation in standard form \( ax^2 + bx + c = 0 \), the quadratic formula can be used to find the x-intercepts:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our example problem, this calculation was applied to \( 2x^2 + 4x - 1 = 0 \), with \( a = 2 \), \( b = 4 \), and \( c = -1 \).
The discriminant part \( \sqrt{b^2 - 4ac} = \sqrt{16 + 8} = \sqrt{24} \) helps to estimate the number and type of roots.
This yields two x-intercepts after solving: \( x = -2.4 \) and \( x = 0.4 \).
These intercepts are useful for sketching the complete graph, showing where the parabola crosses the x-axis.

Graphing a parabola involves plotting several key points derived from the quadratic function and connecting them with a smooth curve. These points typically include:

• The vertex, which provides a central reference point for the shape of the graph.
• The x-intercepts, indicating where the graph crosses the x-axis.
• The y-intercept, found by setting \( x = 0 \), which gives a point where the graph crosses the y-axis.

The parabola opens up or down based on the sign of the coefficient \( a \) in \( ax^2 + bx + c \). If \( a > 0 \), like in our example where \( a = 2 \), the parabola opens upwards. Conversely, if \( a < 0 \), it opens downwards.
After plotting these points, draw a smooth curve that passes through them. Ensure the curve is symmetric around the line \( x = -\frac{b}{2a} \). This symmetry helps maintain the accuracy of the sketch.
In the example, the parabola opens upward, showing a minimum at the vertex \((-1, -3)\), with x-intercepts at \(-2.4, 0.4\) and a y-intercept at \(0, -1\).

The quadratic formula is a powerful algebraic tool used to find the roots of any quadratic equation of the form \( ax^2 + bx + c = 0 \). Regardless of whether the roots are real or complex, the quadratic formula provides solutions systematically.
The formula is stated as:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here:

• \( b^2 - 4ac \) is known as the discriminant.
• The discriminant determines the nature of the roots. If it's positive, there are two distinct real roots. Zero means exactly one real root (a double root), while a negative discriminant indicates complex roots.

In our specific example, the function \( 2x^2 + 4x - 1 \) utilizes this formula to find the roots, yielding real roots \( x = -2.4 \) and \( x = 0.4 \), confirmed by a positive discriminant of 24.
This formula not only aids in finding roots but also informs graph interpretation and geometric properties of the quadratic.