91Ó°ÊÓ

The vertex of a quadratic function is a crucial point on its graph as it represents the highest or lowest point, depending on whether the parabola opens upwards or downwards, respectively. In the given exercise, the vertex was found using the formula
\(x_v = \frac{-b}{2a}\),
where
\(a\) and \(b\) are coefficients from the standard quadratic equation \(ax^2 + bx + c\). After calculating the value of \(x_v\), it's substituted back into the function to find the corresponding \(y\) coordinate. For the example
\(f(x) = x^2 - 10x - 600\),
we can see that the parabola opens upwards, and thus the vertex will be the lowest point on the graph. The calculated coordinates of the vertex are \( (5, -625) \), which helps us understand the function's minimum value and its position on the Cartesian plane.
Knowing the vertex is essential for sketching the graph and recognizing the function's behavior.

The y-intercept of a quadratic function is the point where the graph crosses the y-axis. To find this point for the function \(f(x) = x^2 - 10x - 600\), we set \(x\) to zero and solve for \(f(0)\).
\[f(0) = (0)^2 - 10(0) - 600 = -600\]
The result indicates that the y-intercept occurs at \( (0, -600) \). The significance of the y-intercept lies in its convenience for plotting the graph quickly, as it doesn’t require additional calculations. It's especially useful when sketching the graph by hand, giving a clear starting point to draw the parabola.

The x-intercepts, also known as roots or zeroes of a quadratic function, are the points where the graph intersects the x-axis. These are found by solving \(f(x) = 0\) for \(x\). The exercise provides the step-by-step method to calculate the x-intercepts of the function
\(f(x) = x^2 - 10x - 600\).
Applying the quadratic formula results in two x-intercepts, located at the points:
\( (-20, 0) \) and \( (30, 0) \).
Having both intercepts allows us to understand the symmetry of the parabola around its vertex and provides anchor points to guide us in drawing a precise graph of the quadratic function.

The quadratic formula, a crucial tool for solving quadratic equations, finds the roots of any quadratic function. The generic formula is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
where \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation of the form \(ax^2 + bx + c = 0\). When applied to the provided function
\(f(x) = x^2 - 10x - 600\),
we use the coefficients \(a = 1\), \(b = -10\), and \(c = -600\) to determine the x-intercepts of the graph. This powerful formula delivers the exact points where the parabola will cross the x-axis, as demonstrated by the two solutions in the example, and is versatile enough to be used for any quadratic equation, whether it factors easily or not.