91Ó°ÊÓ

A parabola is a U-shaped curve that can open either upwards or downwards. It is the graph of a quadratic equation of the form \( y = ax^2 + bx + c \). The direction in which a parabola opens is determined by the coefficient \( a \). If \( a \) is positive, the parabola opens upward, and if it's negative, it opens downward. In the equation \( y = x^2 - 2x - 8 \), the coefficient \( a = 1 \) is positive, so this parabola opens upward, resembling a smiling face.
This visual characteristic is crucial because it helps predict how the graph behaves as \( x \) moves away from the vertex. Parabolas are symmetrical, meaning the left side is a mirror image of the right side, which is foundational when graphing.

The vertex of a parabola is its highest or lowest point, serving as a critical plot point. For the quadratic equation \( y = ax^2 + bx + c \), the vertex can be found using a formula for the x-coordinate of the vertex \( h = -\frac{b}{2a} \). After identifying \( h \), you substitute it back into the equation to find \( k \), the y-coordinate.
For the equation \( y = x^2 - 2x - 8 \), \( a = 1 \) and \( b = -2 \). Using the formula, we calculate \( h = 1 \). Substituting \( h \) back into the equation, \( y = 1^2 - 2(1) - 8 \), results in \( k = -9 \). Hence, our vertex is located at \( (1, -9) \).
Understanding how to find the vertex is essential for graphing, providing insight into the parabola's axis of symmetry and its extremity point.

Plotting points is a straightforward way to graph a parabola by calculating y-values for various x-values. Once the vertex is located, as demonstrated earlier, additional points can be calculated by choosing x-values around the vertex, both smaller and larger.
For the equation \( y = x^2 - 2x - 8 \), after finding the vertex at \( (1, -9) \), we calculate additional points by picking \( x = 0 \) and \( x = 2 \). Substituting these into the equation results in points \( (0, -8) \) and \( (2, -8) \).
It's beneficial to choose x-values on both sides of the vertex to ensure symmetry and accuracy in the curve's path. By plotting these points, you prepare for sketching a smooth, continuous curve that forms the parabola.

The quadratic formula is a powerful tool for solving quadratic equations and finding x-intercepts or roots. However, in the context of graphing, it helps gain a deeper understanding of the parabola's behavior.
For a quadratic equation \( ax^2 + bx + c = 0 \), the solutions for \( x \) are given by \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. This formula can identify the x-intercepts by solving \( y = 0 \) for the given quadratic equation.
Although not necessary for graphing, knowing the roots, if they exist, helps graphers understand where the parabola crosses the x-axis, thus broadening the comprehension of the graph's orientation and intercepts. It's a complement to point-plotting, providing a fuller graphical picture.