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Chapter 5: Problem 21

Graph each inequality. $$y \geq x^{2}-9$$

Short Answer

Expert verified
To solve the inequality \(y \geq x^{2}-9\), plot the parabola \(y = x^{2}-9\), then shade the region above and including the parabola.

Step by step solution

01

Plot the quadratic function

Firstly, plot the quadratic function \(y = x^{2}-9\). This is a standard upward-opening parabola, shifted down 9 units on the y-axis.
02

Identify the inequality

The given inequality is \(y \geq x^{2}-9\), which means we are looking for all y-values that are greater than or equal to \(x^{2}-9\). This implies that the required region is above the plotted parabola.
03

Shade the area

Now, shade all the region above the plotted parabola without including the parabola itself. This is done because the inequality symbol '\(\geq\)' points out that the solution involves the function itself \(x^{2}-9\) (the parabola), and every y-value greater.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Parabolas
To graph a parabola, you first need to understand its basic structure. A parabola is a U-shaped curve that results from the graph of a quadratic function. In our example, the quadratic function is \(y = x^{2}-9\). This indicates that our parabola opens upwards (as the coefficient of \(x^2\) is positive) and is shifted down on the y-axis by 9 units.
To plot this parabola, you can start by identifying key points:
  • The vertex, which is the highest or lowest point of the parabola. For \(y = x^{2}-9\), the vertex is at the coordinate (0,-9).
  • The axis of symmetry, which is a vertical line that passes through the vertex. In this case, it is the y-axis, or \(x = 0\).
  • Intercepts - where the parabola crosses the y-axis and any further points for a clearer shape.
Once plotted, the parabola gives a visual representation of the quadratic equation's behavior across the graph.
Inequality Symbols
Inequality symbols can drastically alter the process of graphing a function. For the given exercise, understanding the symbol \( \geq \) is crucial. This symbol stands for "greater than or equal to."
When you see \(y \geq x^{2}-9\), it means you're looking for all y-values which are either on the curve of \(x^2-9\) or higher.
Let's break it down:
  • \( \geq \) includes the values that lie on the line or curve. If it were just \(>\), it would exclude them, and the line or curve would be dotted instead of solid.
  • The solution involves every point satisfying the condition \(y \geq x^{2}-9\), forming an area on the graph above the drawn curve.
  • Use these symbols to determine not only the region you wish to highlight but also whether the boundary is part of the solution.
Shading Regions
Shading regions on a graph helps in visualizing solutions to inequality problems. Once you have graphed the function and identified the inequality, shading becomes a tool to illustrate all the possible solutions. For the inequality \(y \geq x^2 - 9\), you need to shade the region above the parabola's curve.
  • Since \(\geq\) includes the curve itself, start by drawing or bolding the parabola to mark it as part of the solution.
  • Shade all the space that lies above this curve, indicating that all these points meet the inequality requirement.
  • This shading should cover the entire area towards positive infinity on the y-axis, showing any point along this path satisfies \(y \geq x^{2}-9\).
Shading is your visual cue to differentiate regions on the graph, ensuring you can easily interpret the solution set for y-values fulfilling the inequality.

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