91Ó°ÊÓ

The vertex of a parabola is a crucial point that represents the maximum or minimum value of the quadratic function. This point helps define the shape and position of the parabola on the graph. To find the vertex, you need to calculate the x-coordinate using the formula:\[ h = \frac{-b}{2a} \]where \(a\) and \(b\) are coefficients from the standard quadratic equation form \(f(x) = ax^2 + bx + c\).Once we have \(h\), substitute it back into the function to find the y-coordinate \(k\):\[ k = f(h) \]The coordinates \((h, k)\) give you the exact position of the vertex on the Cartesian plane. In our given function \(f(x)=-x^{2}+5x-6\), by substituting \(a = -1\) and \(b = 5\) into the formula, we determine that the x-coordinate \(h\) of the vertex is:\[ h = \frac{-5}{2(-1)} = \frac{5}{2} \]Substituting \(h = \frac{5}{2}\) into \(f(x)\), we find the y-coordinate \(k\):\[ k = f\left(\frac{5}{2}\right) = -\left(\frac{5}{2}\right)^2 + 5\left(\frac{5}{2}\right) - 6 \] which simplifies to:\[ k = \frac{1}{4} \]Thus, the vertex of the parabola is at \(\left(\frac{5}{2}, \frac{1}{4}\right)\).

X-intercepts are points where the graph crosses the x-axis. At these points, the value of the function is zero, meaning:\[ f(x) = 0 \]For a quadratic function \(f(x) = ax^2 + bx + c\), solve the equation \(ax^2 + bx + c = 0\) to find the x-intercepts. You can apply the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]For our function \(f(x) = -x^2 + 5x - 6\), substitute \(a = -1\), \(b = 5\), and \(c = -6\). Compute the discriminant \(b^2 - 4ac\):\[ b^2 - 4ac = 5^2 - 4(-1)(-6) = 25 - 24 = 1 \]Since the discriminant is positive, the parabola crosses the x-axis at two distinct points. Applying the quadratic formula:\[ x = \frac{-5 \pm \sqrt{1}}{-2} \]This simplifies to the x-intercepts:\[ x = 2 \quad \text{and} \quad x = 3 \]These intercepts tell us that the parabola cuts the x-axis at points \((2,0)\) and \((3,0)\).

Graphing parabolas allows us to visualize quadratic functions and understand their behavior in a graphic context. Here are the steps to graph a parabola:

• Find the vertex, as outlined earlier, using the vertex formula.
• Determine the x-intercepts by solving \(f(x) = 0\) to identify where the graph crosses the x-axis.
• Identify other distinctive parameters such as the y-intercept, which is the point \((0,c)\) when \(x=0\).

In our specific case of \(f(x) = -x^2 + 5x - 6\), we established that the vertex is at \(\left(\frac{5}{2}, \frac{1}{4}\right)\), and the x-intercepts are \((2,0)\) and \((3,0)\). The y-intercept occurs at \(f(0) = -6\), giving the point \((0, -6)\).Plot these points on a graph and draw a smooth curve that opens downwards (since \(a = -1 \text{ is negative}\)). Ensure the vertex is the highest point on the graph due to the downward opening.The graph's symmetry about the vertical line through the vertex, known as the axis of symmetry, should be evident. For our equation, this axis is given by \(x = \frac{5}{2}\).Connecting these concepts forms the complete graph of the parabola, showcasing all important features of the quadratic function.