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Understanding how to graph inequalities is important as it visually represents the solutions that satisfy the inequality condition. To begin, we transform the inequality into an equation to outline the boundary of the solutions. In our example, the inequality is \(y \geq x^2 - 1\). By changing the \(\geq\) to \(=\), we get the equation \(y = x^2 - 1\). This forms the boundary line.The boundary line or curve helps us determine which region to shade.

• If the inequality symbol is \(<\) or \(\leq\), the shaded area will be below the boundary.
• If it is \(>\) or \(\geq\), as in our case, you will shade above the boundary.

Remember to include the boundary line/curve if the inequality sign is \(\leq\) or \(\geq\), indicating that points on the line are solutions too, which means using a solid line. Otherwise, use a dashed line.

A parabola is a type of graph that is symmetric and has a characteristic U-shape. Parabolas are formed by quadratic equations, which are polynomial equations of degree two. The standard form for a quadratic equation is \(y = ax^2 + bx + c\), and each parabola will open upwards or downwards. The way the parabola opens is determined by the sign of the \(a\) coefficient:

• If \(a\) is positive, it opens upwards.
• If \(a\) is negative, it opens downwards.

The vertex of the parabola is its highest or lowest point, depending on the direction it opens. The parabola in our example, \(y = x^2 - 1\), opens upwards since the coefficient of \(x^2\) is positive. It has its vertex at (0, -1). Understanding the vertex helps identify the axis of symmetry, which is a vertical line passing through the vertex, ensuring the left and right sides of the parabola mirror each other.

The vertex form of a quadratic equation is a convenient way to represent parabolas. It is written as \(y = a(x - h)^2 + k\), where \((h, k)\) are the coordinates of the vertex.This form is particularly helpful because it easily indicates:

• The vertex, \((h, k)\), shows the peak or minimum of the parabola.
• The value of \(a\) tells us whether the parabola opens upwards or downwards.

By rewriting our equation \(y = x^2 - 1\) in vertex form, it becomes clear that it still equals \(y = 1(x - 0)^2 - 1\), simplifying to \(y = x^2 - 1\). The vertex is at \((0, -1)\), confirming how this form directly points to the key features of the parabola.Using the vertex form transforms the process of graphing and analyzing parabolas into a straightforward, less error-prone task.