Problem 38 - Precalculus with Limits

Question

In Exercises 33-38, (a) use a graphing utility to graph the function and find the zeros of the function and (b) verify your results from part (a) algebraically. $f(x) = {2x^2 - 9}{3-x}

Step-by-Step Solution

Answer

The zeros of the function \(f(x) = \frac{{2x^2 - 9}}{{3-x}}\) can be determined by graphing the function and then verifying these zeros algebraically. You can first graph the function, note the x-coordinates where the function intersects the x-axis and then substitute these values back into the function to verify that they are indeed zeros.

Step 1: Graphing the function

The given function \(f(x) = \frac{{2x^2 - 9}}{{3-x}}\) should be graphed using a graphing utility. The x-intercepts on the graph indicate the zeros of the function. Make a note of the x-coordinates of these points.

Step 2: Analyzing the function graphically

Scan the graph produced by the utility for any points where the curve intersects the x-axis. These points indicate the zeros of the function. Make sure to accurately identify these points.

Step 3: Verifying algebraically

After the zeros are found graphically, substitute these values into the given function to see if the equation equals to zero. Equation for zero is \(f(x) = 0\), which means \((2x^2 - 9) = 0\) (Given \(3 - x \neq 0\)) or \(3 - x = 0\) (Given \(2x^2 - 9 \neq 0\)). Solve these equations to find the x values and compare with the zeros obtained graphically.

Key Concepts

Graphing Utility

When dealing with complex functions, a graphing utility can be your best friend. These tools are designed to visually represent mathematical functions, which can help you understand their behavior without solving complicated equations by hand. A graphing utility plots the graph of the function and clearly shows where the curve intersects the x-axis. These intersection points are known as the zeros (or roots) of the function. They tell us exactly where the function's output is zero.

In the given function, using a graphing utility allows us to quickly see where the curve crosses the x-axis. This graphical representation can uncover zeros that might be obscure when tackling the equation algebraically. Keep in mind:

• Ensure that your graphing utility is correctly set to display the relevant range.
• Check the precision of your graphing tool to identify zeros more accurately.
• After identifying these zeros, it's crucial to verify them using algebraic methods for confirmation.

Algebraic Verification

While graphing utilities provide a visual insight into the zeros of a function, algebraic verification solidifies those findings by relying on mathematical principles. After identifying the zero points graphically, you should check them by plugging these x-values back into the original function. This ensures that substituting these values truly equals zero, validating their correctness.

To perform algebraic verification on the function \[f(x) = \frac{{2x^2 - 9}}{{3-x}},\]you should set the numerator and denominator separately to zero and solve each.

• Solve \(2x^2 - 9 = 0\) to find potential zeros from the numerator.
• Solve \(3 - x = 0\) and consider conditions to ensure the denominator never equals zero to avoid undefined results.

This double-checking method helps cross-reference the x-values obtained from the graphing utility to ensure that they are correct and hold mathematically sound solutions.

Quadratic Functions

Understanding the nature of quadratic functions is key when finding zeros. A quadratic function typically has the form \(ax^2 + bx + c\), and its graph is a parabola. The vertex form and standard form both provide insights into the function's behavior, such as direction and where it crosses the x-axis. In our original function, \[\frac{{2x^2 - 9}}{{3-x}},\]the numerator represents a quadratic function, \(2x^2 - 9\).

Let's break it down:

• The coefficient in front of \(x^2\) (here, \(a = 2\)) tells you the parabola's direction; positive means it opens upwards.
• Zeros of this quadratic are found by setting it to zero and solving: \(2x^2 - 9 = 0\). This simplifies to finding when \(x^2 = \frac{9}{2}\).
• Real solutions (if they exist) are the places where the graph crosses the x-axis. These are the values we initially found using the graphing utility.

Grasping these concepts allows a deeper understanding of when and why a quadratic function has certain zero points, enhancing both graphical and algebraic problem-solving skills.