91Ó°ÊÓ

A parabola is a symmetric, U-shaped curve that is defined by a quadratic equation. In the inequality given in our exercise, we explore the parabola defined by the equation \( y = x^2 + 1 \). Here, the graph of the parabola serves as a boundary for the inequality \( y > x^2 + 1 \). To understand a parabola, it's critical to identify the vertex, which is the point where the parabola changes direction. In simpler terms, the parabola looks like a cup if it opens upwards or an upside-down cup if it opens downwards. In our example, the parabola opens upward because the coefficient of \( x^2 \) is positive. Points to remember about a parabola:

• The basic form is \( y = ax^2 + bx + c \).
• The vertex is derived from the terms \( a \), \( b \), and \( c \).
• A positive \( a \) means the parabola opens upwards.

When sketching, calculate and plot a few key points around the vertex for accuracy.

The vertex form of a quadratic equation is highly useful for graphing parabolas as it reveals the vertex directly. The standard vertex form is expressed as \( y = a(x - h)^2 + k \), where

• \( (h, k) \) is the vertex of the parabola, and
• \( a \) determines the direction and the width of the parabola.

In our exercise, even though the equation \( y = x^2 + 1 \) does not look like the classic vertex form, it can also be seen as \( y = (x - 0)^2 + 1 \), identifying the vertex as \((0, 1)\). This helps to quickly spot where the parabola "sits" on the graph. By knowing these parameters,

• The vertex provides the starting point for drawing the curve,
• \( a \) confirms the parabola's direction (upward or downward), and
• \( c \) gives the vertical shift from the base parabola \( y = x^2 \).

When graphing inequalities, shading the correct region is key to representing the solution set visually. For the inequality \( y > x^2 + 1 \), the process involves determining which part of the graph satisfies the inequality. Once the parabola is sketched using a dashed line, as the inequality is strict (indicated by \(>\) rather than \( \geq \)), shading takes place above this boundary line. Steps for shading:

• Identify the boundary line or curve (here, the parabola \( y = x^2 + 1 \)).
• Since the inequality is strict, use a dashed line to show the boundary isn’t included.
• Determine which region satisfies the inequality by choosing a test point, such as the origin \((0, 0)\). Substitute it into the inequality: \( y > x^2 + 1 \).
• If the test fails (like in this case \(0 > 1\) is false), the region that satisfies the test is on the opposite side of the test point.

Correct shading visually represents every possibility that meets the inequality’s condition, helping us "see" the solution.