Problem 24 - Algebra 1

Question

Sketch the graph of the inequality. $$y \geq x^{2}-3$$

Step-by-Step Solution

Answer

The graph of the inequality \(y \geq x^{2}-3\) is a parabola opening upwards and situated 3 units below the y-axis. The solution region to the inequality, being \(y \geq x^{2}-3\), is the area above and on the curve of the parabola, including the parabola itself.

Step 1: Sketch the parabola

Draw the graph of the quadratic equation \(y = x^{2}-3\). This is a simple parabola shifted 3 units down the y-axis, with the vertex at (0, -3). Make sure the parabola opens upwards.

Step 2: Identify the inequality

Notice that the inequality is \(y \geq x^{2}-3\). This means that we're interested in the areas at and above the curve. The parabola divides the plane into two regions, the region inside the curve 'below' and the region outside the curve 'above'.

Step 3: Shade the solution region

As the given inequality is \(y \geq x^{2}-3\), this indicates the solution lies above or on the curve. So, shade the area above and on the parabola to represent the solution of the inequality.

Key Concepts

Quadratic Functions

Quadratic functions are a special form of polynomial functions where the highest degree of the variable is 2. They are commonly written in the standard form: \[y = ax^2 + bx + c\]where:

• \(a\), \(b\), and \(c\) are constants
• \(a eq 0\) since if \(a = 0\), the equation becomes linear

Quadratic functions are highly impactful in mathematics due to their simplicity and the visual patterns they form, particularly the shape of their graphs, which are parabolas. These functions can model various real-world phenomena, such as projectile motion. Understanding the basic properties and transformations of quadratic functions is essential for graphing and interpreting inequalities involving such functions. For instance, when graphing an inequality like \(y \geq x^2 - 3\), we first focus on the quadratic part \(x^2 - 3\), which helps determine the parabola’s shape and position.

Parabolas

The graph of a quadratic function is a curve called a parabola. Parabolas have a distinct 'U' shape and come in two orientations:

• Opening upwards, if the coefficient \(a > 0\)
• Opening downwards, if \(a < 0\)

For the quadratic equation \(y = x^2 - 3\), the parabola is oriented upwards because the coefficient of \(x^2\) is positive.

Key features of a parabola include:

• The **vertex**: The parabola's turning point; here, it is at (0, -3).
• The **axis of symmetry**: A vertical line that runs through the vertex, here it is the line \(x = 0\).
• The **direction of opening**: Upwards, indicating the parabola gets wider as you move away from the vertex along the x-axis.

Graphs of inequalities involving parabolas use these features to determine which part of the plane satisfies the inequality. Understanding how to sketch and interpret these graphs is vital for solving quadratic inequalities.

Shading Regions

Shading regions on a graph is a technique used to visually represent the solutions of inequalities. When working with quadratic functions, shading involves determining which area relative to the parabola satisfies the inequality.

For example, with the inequality \(y \geq x^2 - 3\), the solution set includes all points on the parabola \(y = x^2 - 3\) and those above it. This is because the inequality specifies \(y\) values "greater than or equal to" the quadratic expression.

To correctly shade:

• Identify the boundary curve, here it is \(y = x^2 - 3\).
• Since the inequality includes equality (\(\geq\)), the boundary should be solid, not dashed.
• Shade the region above the parabola, as it represents all \(y\) values greater than the curve.

This shading technique is crucial for graphing solutions to inequalities—ensuring clear visual identification of where solutions lie.