91Ó°ÊÓ

The vertex of a parabola is a crucial point, where the curve changes direction. This point is either the highest or lowest on the parabola, depending on the direction in which it faces.
For a parabola given by the quadratic equation \( y = ax^2 + bx + c \), the vertex can be found using the formula for the x-coordinate:

• \( x = -\frac{b}{2a} \)

In this scenario, substitute the values from the equation \( y = -2x^2 + 6x - 3 \) into the formula:

• \( a = -2 \)
• \( b = 6 \)

Calculate \( x \): \[ x = -\frac{6}{2(-2)} = \frac{6}{4} = \frac{3}{2} \]
Once the x-coordinate is found, substitute it back into the original equation to find the y-coordinate. \[ y = -2\left(\frac{3}{2}\right)^2 + 6\left(\frac{3}{2}\right) - 3 = \frac{3}{2} \]
Thus, the vertex of the parabola is \((\frac{3}{2}, \frac{3}{2})\). This specific parabola is downward-facing, as indicated by the negative \(a\) value.

The y-intercept of a graph is the point where it crosses the y-axis. This means the value of \( x \) is zero at this point. To find the y-intercept for a parabola represented by the equation \( y = ax^2 + bx + c \), simply set \( x = 0 \) and solve for \( y \).
In the equation \( y = -2x^2 + 6x - 3 \), plug in \( x = 0 \):

• \( y = -2(0)^2 + 6(0) - 3 = -3 \)

So, the y-intercept is \((0, -3)\). This means the parabola crosses the y-axis at the point \((0, -3)\), which is a valuable piece of information when sketching the graph.

The x-intercepts of a parabola are the points where the graph crosses the x-axis, implying that \( y = 0 \) at these points. Finding these intercepts involves setting the equation equal to zero and solving for \( x \).
For our equation \( -2x^2 + 6x - 3 = 0 \), we'll use the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Calculate the discriminant \( b^2 - 4ac \), which indicates the nature of the roots:

• \( b = 6 \), \( a = -2 \), \( c = -3 \)
• Discriminant: \( 6^2 - 4(-2)(-3) = 36 - 24 = 12 \)

Since the discriminant is positive, there are two real solutions, meaning the parabola intersects the x-axis at two points.Calculate the x-intercepts:\[ x = \frac{-6 \pm \sqrt{12}}{-4} \]\[ = \frac{6 \pm 2\sqrt{3}}{4} \]\[ = \frac{3 \pm \sqrt{3}}{2} \]
Thus, the x-intercepts are \((\frac{3 + \sqrt{3}}{2}, 0)\) and \((\frac{3 - \sqrt{3}}{2}, 0)\). Recognizing these intercepts helps in understanding where the graph of the parabola touches the x-axis.

The quadratic formula is a powerful tool for finding solutions (roots) of quadratic equations. A standard quadratic equation is expressed as \( ax^2 + bx + c = 0 \). The formula provides solutions for \( x \) by computing:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula is derived from completing the square of the quadratic equation and is universally applicable for finding real or complex roots. The discriminant, \( b^2 - 4ac \), plays a critical role in determining the nature of the roots:

• If \( b^2 - 4ac > 0 \), there are two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is one repeated real root.
• If \( b^2 - 4ac < 0 \), the roots are complex (not real).

In our example with the equation \( -2x^2 + 6x - 3 = 0 \), we calculated the discriminant to be 12, which is greater than zero. This confirms two real solutions, aiding in finding the x-intercepts of the given parabola. By following this method, you can solve any quadratic equation with ease.