91Ó°ÊÓ

Understanding the standard form of a quadratic equation is the first step in identifying and analyzing parabolas. The standard form is expressed as \(y = ax^2 + bx + c\). Here,

• \(a\) is the coefficient of \(x^2\), which influences the "width" and "direction" of the parabola (whether it opens upwards or downwards).
• \(b\) is the coefficient of \(x\), impacting the "location" of the parabola along the x-axis.
• \(c\) is the constant term, determining the "y-intercept," the point at which the parabola crosses the y-axis.

In the problem, the equation is \(y = x^2 + 4x - 5\). By identifying \(a = 1\), \(b = 4\), and \(c = -5\), we gather essential details to further work on finding the vertex and graphically representing the parabola.

The vertex of a parabola is a crucial feature that indicates its highest or lowest point. For any quadratic equation in standard form \(y = ax^2 + bx + c\), the x-coordinate of the vertex can be efficiently calculated using the formula \(x = -\frac{b}{2a}\). To find the vertex for our given equation:1. Identify \(a\), \(b\), and \(c\).2. Plug these values into the formula to determine the x-coordinate.In this case, \(a = 1\) and \(b = 4\), leading to \(x = -\frac{4}{2 \cdot 1} = -2\).To find the y-coordinate, substitute \(x = -2\) back into the quadratic equation:\[y = (-2)^2 + 4(-2) - 5 = 4 - 8 - 5 = -9\]Thus, the vertex of the parabola for this equation is \((-2, -9)\). Understanding this step significantly aids in visualizing the shape and position of the parabola.

Graphing a parabola involves plotting its key components accurately on a coordinate plane. Start by marking the vertex, which in this case is \((-2, -9)\). The direction the parabola opens—that is, whether it curves up or down—depends on the coefficient \(a\). When \(a > 0\), as with \(y = x^2 + 4x - 5\), the parabola opens upward. To plot the parabola:

• Start by plotting the vertex point, establishing the parabola's central axis.
• Choose additional points on each side of the vertex to show the parabola's shape. Substituting x-values into the equation gives corresponding y-values.
• Draw a smooth curve through these points, ensuring the curve is symmetric about the vertex's vertical line.

This systematic approach not only helps in constructing an accurate visual of the parabola but also reinforces understanding of how changes in \(a\), \(b\), and \(c\) affect its graph.