Question

Use the zoom and trace features of a graphing utility to approximate the real zeros of \(f\). Give your approximations to the nearest thousandth. $$f(x)=x^{3}-3.9 x^{2}+4.79 x-1.881$$

Short answer

The real zeros will vary based on the specifics of each approach and graphing utility. So, double-check those zeros using the trace feature again after zooming in. The answer should be left in decimal and rounded to the nearest thousandth.

Step-by-step solution

Plot the Function f(x)

Use a graphing utility to plot the function \(f(x) = x^{3}-3.9 x^{2}+4.79x-1.881\). The curve of the graph will help visualize the zeros of the function.

Identify Potential Zeros Using the Trace Feature

Use the trace feature of your graphing utility. The trace feature moves a cursor along the graph of the function and provides the x and y coordinates of each position. As you trace, look for x-values where the function intersects the x-axis (i.e., y=0), as these are the zeros of the function.

Zoom in on Potential Zeros

Each point where the function intersects the x-axis is a potential zero. These points would typically be where the y-coordinate of the function is 0. Use the zoom feature of your graphing utility to zoom in at these points to get a more accurate x-value for each one.

Approximate the Real Zeros

After zooming in, use the trace feature again to get the x-values where y is approximately 0 to the nearest thousandth. These x-values are the real zeros of your function.

Key concepts

Real Zeros

Finding real zeros is crucial for understanding the behavior of polynomial functions. A real zero is an x-value where the graph of a function intersects the x-axis. At these points, the function has a value of zero, meaning it solves the equation \( f(x) = 0 \).
To approximate real zeros with a graphing calculator, you can visually identify these intersection points. The task can be challenging because you may not always see the precise point where the function crosses the x-axis just by looking at the graph. That's why tools like the 'trace' and 'zoom' features are very helpful.

• Tracing helps you to navigate along the curve and observe the rough areas where an intersection appears.
• Zooming allows you to take a closer look and pinpoint where the value of y approaches zero.

By combining these features, you can find these approximations more accurately, sometimes necessary for complex polynomial functions where manual calculations are cumbersome.

Trace Feature

The trace feature in a graphing calculator is an extremely helpful tool for analyzing graphs. When using the trace feature, you can move a cursor along the plotted curve. The calculator will display the coordinates of the cursor position.
This is particularly useful for identifying key points such as real zeros or turning points. For our polynomial \( f(x) \), the trace feature lets you identify approximate x-values where the function seems to cross the x-axis.

• Navigate easily: Using arrow keys, you can smoothly trace through the graph.
• Spotting zeros: As you watch the y-coordinates change, look for points where they get closer to zero.

By keeping a close watch on changes in coordinate values, especially y-values nearing zero, you can estimate the real zeros effectively. This feature is essential for honing in on specific values that are important for more in-depth analysis.

Zoom Feature

The zoom feature is another powerful aspect of graphing calculators. This feature allows you to magnify or zoom into a particular part of the graph to observe it in greater detail. For functions like \( f(x) = x^{3}-3.9 x^{2}+4.79 x-1.881 \), where precise calculation of zeros is needed, zooming is indispensable.

• Enhances precision: By zooming in, you can refine the estimation of zeros further than what a standard view might offer.
• Focuses on intersections: Helps to clearly see where the curve meets the x-axis, enabling better geometric insight.

Using the zoom feature after initially using trace helps by providing a closer look at subtle changes on the graph. After zooming, using the trace again can lead to refined x-value approximations, bringing you closer to exact real zero values which are needed for computations to the nearest thousandth.

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. This specific type of function is foundational in algebra and calculus due to its broad application. For the given polynomial \( f(x) = x^{3}-3.9 x^{2}+4.79 x-1.881 \), understanding its real zeros helps in revealing its roots or solutions.

• Degree relevance: The degree of the polynomial (in this case, 3) tells us about the maximum number of potential real zeros.
• Graph behavior: The nature of the curve, with its hills and valleys, can indicate where these zeros may lie.

Exploring polynomial functions with graphing calculators enables effective visualization. It simplifies the process of identifying significant features such as zeros, turning points, and end behavior. This understanding is vital for solving equations and predicting future values of these mathematical models in practical applications.