Problem 89 - Algebra and Trigonometry Real Mathematics, Real People

Question

Graph the function and determine the interval(s) (if any) on the real axis for which \(f(x) \geq 0\) Use a graphing utility to verify your results. $$f(x)=x^{2}-9$$

Step-by-Step Solution

Answer

The intervals on the real axis for which \(f(x) \geq 0\) are \(-\infty , -3] \cup [3, \infty)\)

Step 1: Graphing the function

The function \(f(x) = x^{2} - 9\) can be graphed as a parabola opening upwards with a vertex at the point (0, -9) and x-intercepts at \(x = -3\) and \(x = 3\). This is because \(f(x) = 0\) when \(x^{2} - 9 = 0\), resulting in \(x = -3, 3\). In conclusion, graphing helps visualize where \(f(x) \geq 0\). If we set \(f(x) = x^{2}-9\) to be greater than or equal to 0, the solutions to this inequality will be the x-values where the function lies above the x-axis (inclusive).

Step 2: Solve the inequality

Find the intervals for which \(f(x) \geq 0\) by solving the inequality \(x^{2} - 9 \geq 0\). From the graph, it can be observed that for \(x^{2} - 9\) to be greater than or equal to 0, \(x\) must be less than or equal to -3 or greater than or equal to 3. So the intervals on the real axis for which \(f(x) \geq 0\) are \(-\infty , -3] \cup [3, \infty)\)

Step 3: Verification using a graphing utility

After working out the intervals manually, confirm the result using a graphing utility such as Desmos or GeoGebra. Plotting the function in the graphing tool, the region of the graph above the x-axis should correspond to the intervals \(-\infty , -3] \cup [3, \infty)\), which matches the manually worked out intervals.

Key Concepts

Inequalities in Algebra

When studying algebra, understanding inequalities is crucial for solving a wide range of mathematical problems. An inequality signals that two values are not equal and specifies the relation between them. They come in several types, including 'less than' (<), 'greater than' (>), 'less than or equal to' (≤), and 'greater than or equal to' (≥).

In the context of graphing quadratic functions like \( f(x) = x^{2} - 9 \), inequalities help us determine where the function's value is either above or below a certain level. When solving the inequality \( f(x) \textgeq 0 \), you're essentially looking for where the parabola sits above the x-axis (where y is non-negative). This translates to finding the x-values that satisfy the condition of the inequality. In our example, it's evident that this is the case when \( x \textleq -3 \) or \( x \textgeq 3 \), creating intervals on the real axis.

Parabola Graph Characteristics

Understanding the characteristics of the parabola graph is a foundational concept when dealing with quadratic functions. Graphs of these functions create a U-shaped curve called a parabola, which can open upwards or downwards.

For the quadratic function \( f(x) = x^{2} - 9 \), the parabola opens upwards because the coefficient of \( x^{2} \) is positive. The most critical points to identify on the graph are the vertex and the x-intercepts. The vertex is the point at which the parabola changes direction. In this example, the vertex is at (0, -9).

The x-intercepts, also known as the roots or zeros, are the points where the parabola crosses the x-axis. For \( f(x) = x^{2} - 9 \), the x-intercepts are at \( x = -3 \) and \( x = 3 \), as these values 'zero out' the function. Additionally, the symmetry of the graph means that if \( f(a) = f(b) \), then the vertical line passing through the midpoint of a and b (the axis of symmetry) bisects the parabola into mirror images.

Using Graphing Utilities

Graphing utilities like Desmos or GeoGebra offer a powerful tool for visualizing algebraic concepts, especially when graphing quadratic functions. These tools allow users to see the shape and position of the parabola instantly, providing immediate visual feedback on the function's characteristics.

To verify solutions or enhance understanding, input the quadratic equation into a graphing utility. The area of the graph that corresponds to \( f(x) \textgeq 0 \) will show as the portion of the parabola that is at or above the x-axis. This visual representation supports the algebraic solution found when solving the inequality by hand. When you use such utilities, pay attention to key features like the vertex, axis of symmetry, and x-intercepts for a comprehensive analysis of quadratic functions.

These tools can cement understanding by confirming the intervals on the real axis where the function's value is non-negative, which, for \( f(x) = x^{2} - 9 \), are \( -\textinfinity, -3] \textcup [3, \textinfinity) \). They also offer the opportunity to explore the effects of different parameters on the shape and location of the parabola, making them indispensable for both intuitive learning and precise graphing.