91Ó°ÊÓ

The vertex of a parabola provides valuable information about the graph's turning point. In the standard quadratic equation form, \(y = ax^2 + bx + c\), the vertex can be found at the coordinates \((h, k)\). To determine \(h\), use the formula \(h = -\frac{b}{2a}\). In the given equation \(y = x^2 - 4\), comparing it with the standard form tells us that \(a = 1\), \(b = 0\), and \(c = -4\). Because \(b\) is zero, \(h\) simplifies to \(h = 0\) after calculating \(-\frac{0}{2 \cdot 1}\). The \(k\) value is directly the \(c\) term, resulting in a vertex at the point \((0, -4)\). The vertex being \((0, -4)\) means it's also the lowest point on the parabola because the parabola opens upwards (as \(a > 0\)). This critical point dictates much of the graph's shape and is a starting point for graphing the parabola successfully.

Finding the x-intercepts is an essential part of graphing a parabola. These points are where the graph of the parabola crosses the x-axis and therefore occur where \(y = 0\). To find the x-intercepts of the quadratic equation \(y = x^2 - 4\), set the equation equal to zero: \(x^2 - 4 = 0\). This can be rewritten by factoring it as \((x - 2)(x + 2) = 0\) .
Setting each factor equal to zero provides two solutions:

• \(x - 2 = 0\) gives \(x = 2\)
• \(x + 2 = 0\) gives \(x = -2\)

These solutions correspond to the intercepts \((2, 0)\) and \((-2, 0)\). Plotting these intercepts helps form the parabolic shape by marking the points where the parabola crosses the x-axis.

The y-intercept is the point where the parabola touches the y-axis, occurring when \(x = 0\). For the equation \(y = x^2 - 4\), substituting \(x = 0\) gives the calculation: \(y = 0^2 - 4 = -4\)This results in the y-intercept located at the point \((0, -4)\). Interestingly, this point is exactly where the vertex is located, revealing symmetry about the y-axis of this particular parabola. Graphically, the y-intercept helps anchor the parabola on the graph and contributes to understanding how the parabola is positioned in relation to other plotted points.

Using a graphing calculator can greatly help visualize the function \(y = x^2 - 4\). This tool allows for precise graph plotting and verification of calculated points, ensuring an accurate representation of the parabola.To use a graphing calculator effectively,

• Input the quadratic equation into the function plotter to generate the graph.
• Locate the vertex, x-intercepts, and y-intercept visually on the display, confirming their positions match predicted values.
• Utilize the zoom and trace features to closely inspect portions of the graph or to measure distances.

Graphing calculators enhance understanding by providing immediate, visual feedback on the parabolic shape, its symmetry, and the intercepts, assisting learners in grasping how algebraic manipulations appear graphically.