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The vertex of a parabola is the point where the graph changes direction. It can be the highest or lowest point on the graph, depending on the parabola's orientation. For a parabola in the form, \( f(x) = ax^2 + bx + c \), the vertex can be found using a specific formula for the x-coordinate of the vertex:

• \( x_{vertex} = \frac{-b}{2a} \)

This formula uses the coefficients of the quadratic expression, where \( a \) is the coefficient of \( x^2 \), and \( b \) is the coefficient of \( x \). Once you calculate \( x_{vertex} \), plug it back into the equation to find the y-coordinate of the vertex:

• \( y_{vertex} = f(x_{vertex}) \)

This gives the vertex as the point \((x_{vertex}, y_{vertex})\). For example, in the function \( f(x) = -2x^2 + 6x - 3 \), the vertex calculated is \( (\frac{3}{2}, -\frac{3}{2}) \). This vertex is crucial because it tells us both the direction of the graph and its extremum point.

The quadratic formula is a powerful tool used to find the roots of a quadratic equation, which are the x-values at which the graph of the equation crosses the x-axis. A quadratic equation is generally expressed in the form \( ax^2 + bx + c = 0 \). The quadratic formula provides a direct solution for \( x \):

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

The term under the square root sign, \( b^2 - 4ac \), is called the discriminant. It determines the nature of the roots:

• If \( b^2 - 4ac > 0 \), there are two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is one real root (also called a repeated or double root).
• If \( b^2 - 4ac < 0 \), there are no real roots but rather two complex roots.

For the function \( f(x) = -2x^2 + 6x - 3 \), using the quadratic formula gives the solutions for the x-intercepts: \( x = \frac{-6 \pm \sqrt{12}}{-4} \). This calculation is fundamental for determining how the parabola intersects the x-axis.

X-intercepts, also known as roots or zeros, are the points where a graph crosses the x-axis. For any function, finding the x-intercepts involves determining the x-values for which the function equals zero. For a quadratic function \( f(x) = ax^2 + bx + c \), you set \( f(x) = 0 \) and solve for \( x \).The x-intercepts can give vital information about the parabola's behavior and can be found using factoring (if possible), completing the square, or more commonly, the quadratic formula as described previously. For the quadratic \( f(x) = -2x^2 + 6x - 3 \), the x-intercepts are the solutions \( x = \frac{-6 + \sqrt{12}}{-4} \) and \( x = \frac{-6 - \sqrt{12}}{-4} \). These tell us where the parabola touches or crosses the x-axis.Knowing the position of x-intercepts within the graph helps visualize and sketch the shape of the parabola effectively, helping us understand the function's practical applications.