91Ó°ÊÓ

A parabola is a symmetric curve that you often encounter in mathematics. Parabolas can open upwards, downwards, to the left, or to the right. In this exercise, we deal with a standard upward-opening parabola. This specific parabola represents the equation \(y = x^2 - 3\).

Important properties of parabolas include:

• Shape: U-shaped, with a point known as the vertex.
• Axis of Symmetry: The vertical line that runs through the vertex, dividing the parabola into two mirror-image halves.
• Focus and Directrix: Parabolas have a point called the focus and a line called the directrix that help define the curvature.

The equation \(y = x^2 - 3\) reveals that this parabola shifts the standard \(y = x^2\) down by 3 units, which affects its position on the coordinate plane.

When dealing with inequalities like \(y > x^2 - 3\), the solution set includes all the points where the inequality holds true. This isn't just one line or curve, but a region on the graph.

Key insights about solution sets in the context of inequalities include:

• A solution set for \(y > x^2 - 3\) contains all the points located above the parabola \(y = x^2 - 3\).
• The boundary of the solution is the parabola itself, graphed as a dashed line to signify points on it are not included.
• Solution sets can be verified by testing random points within the shaded area to confirm they satisfy the inequality.

In this example, the test point \((0, 0)\) supports that the solution region is everything above the curve.

The coordinate plane is the dual-axis system that graphs to visually represent functions and equations. It consists of a horizontal line called the x-axis and a vertical line called the y-axis. These axes intersect at a point called the origin \(0,0\), which acts as the center point.

When graphing an equation, the coordinate plane allows for a clear depiction of:

• Position: Each point is defined by an \((x, y)\) coordinate that indicates its horizontal and vertical locations.
• Graphs: Equations like \(y = x^2 - 3\) are drawn on this plane, facilitating visual analysis of their behavior.
• Quadrants: The plane is divided into four quadrants that help contextualize where a point or graph segment lies.'

In this exercise, we use the coordinate plane to graph the inequality and determine the appropriate shaded region.

The vertex of a parabola is its highest or lowest point, depending on the direction it opens. For the upward-opening parabola \(y = x^2 - 3\), the vertex is the lowest point.

To find the vertex in equations of the form \(y = ax^2 + bx + c\), use the formula
\[ x = -\frac{b}{2a}\]Since our equation is \(y = x^2 - 3\), we have \(a = 1\), \(b = 0\), and \(c = -3\). Substituting these into our formula, we have
\[ x = -0/(2 \times 1) = 0\]Then, plugging \(x = 0\) back into the equation for y gives you the vertex's coordinates:
\[ y = (0)^2 - 3 = -3\] Thus, the vertex is at \((0, -3)\). This point helps position the parabola accurately on the graph and is crucial for determining the boundary for inequalities.