Problem 22 - College Algebra

Question

In Exercises 1–26, graph each inequality. $$y \geq x^{2}-1$$

Step-by-Step Solution

Answer

The graph of the inequality \(y \geq x^{2}-1\) is an upward-opening parabola with the vertex at (0,-1), and the region above the parabola (including the line itself) shaded.

Step 1: Plotting the function

Start by plotting the function \(y = x^{2}-1\). This is a standard parabola opening upwards, but shifted down by 1 unit along the y-axis. So, the vertex of the parabola is at (0,-1).

Step 2: Shading the inequality

The inequality is \(y \geq x^{2}-1\). This means that 'y' can be any value larger than or equal to the function \(x^{2}-1\). Hence, shade the region above the curve y=x^2-1 including the line itself.

Step 3: Final Graph

Review your graph to make sure you have correctly plotted the parabola and shaded the necessary region. The graph of the function should be a solid line (because the inequality includes the equal sign) and the area above it should be shaded.

Key Concepts

Parabola

A parabola is a U-shaped curve that can open up or down. It is the graph of a quadratic function, which generally takes the form:

• \( y = ax^2 + bx + c \)

In our exercise, we have the function \( y = x^2 - 1 \), which is a common example of a parabola. Here, the coefficient of \( x^2 \) is positive, meaning the parabola opens upwards.

This specific parabola is centered at the origin, but is shifted down by 1 unit. The vertex, the lowest point of this upward-opening parabola, is located at (0, -1).

• To plot the parabola, identify key points such as the vertex, and use the symmetry of the parabola to find additional points.
• Since there are no linear or constant terms beyond the vertical shift, the parabola is symmetric about the vertical axis.

When graphing such functions, the understanding of the parabola's vertical and horizontal shifts helps establish a precise sketch of the function's curve.

Shading Solutions

When dealing with inequalities, the shaded region on a graph represents all the possible solutions. In our inequality \( y \geq x^2 - 1 \), we are interested in all points where y is greater than or equal to the quadratic function \( y = x^2 - 1 \).
The outline of the shading process is as follows:

• First, graph the boundary given by \( y = x^2 - 1 \). This curve acts as the border for the region where solutions are located.
• Since the inequality sign is 'greater than or equal to' \( \geq \), use a solid line to show the boundary. This indicates that points on the parabola are included in the solution set.
• Then, shade the entire region above this line. Shading above the curve represents all points \( (x, y) \) where y is greater than those on the curve.

By following these steps, you create a visual map of where all the solutions to the inequality exist.

Quadratic Function

Quadratic functions are extremely important in algebra and calculus, and understanding their properties deeply enhances your overall mathematical insight. These functions are defined as any function that can be expressed in the form:

• \( y = ax^2 + bx + c \)

In the context of our problem, \( y = x^2 - 1 \) is a quadratic function. Here, the term \( a = 1 \) indicates a standard parabola, \( b = 0 \) means there's no linear component, and \( c = -1 \) shifts the graph downward.

• Quadratic equations offer a symmetrical structure, and their graphs are always parabolas.
• They are characterized by a single vertex, and may have up to two x-intercepts depending on how the parabola is positioned relative to the x-axis.

Understanding how each coefficient, \( a, b, \) and \( c \), affects the graph will enable you to predict and sketch the graph accurately. Thus, quadratic functions not only define parabola shapes but also provide a framework for exploring more complex algebraic solutions.