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. $$f(x)=-x^{3}+2 x^{2}+4 x+5$$

Short answer

The real zeros of \(f(x)=-x^{3}+2 x^{2}+4 x+5\) can only be approximated through the use of a graphing utility. Each one varies based on the graphical representation and the precision of tracing.

Step-by-step solution

Plot the Function f(x)

Enter the function \(f(x)=-x^{3}+2 x^{2}+4 x+5\) into the graphing utility. Set the appropriate window size to get a complete view of the function. This allows to identify the regions where the function intercepts the x-axis.

Use the Zoom Feature

Next, use the zoom feature of the graphing utility to zoom in the areas where the function f(x) is crossing the x-axis. This is to get an accurate position for the function's intercepts.

Use the Trace Feature

After identifying the regions of interception, utilize the trace feature of the utility. This feature enables to move along the graph, providing the (x, y) coordinates at each point. Track the x-values where the function intercepts the x-axis, these x-values provide the real zeros.

Approximate the Real Zeros

Given the requirement to express the real zeros to the nearest thousandth, round off the traced values (intercepts) to three decimal places. The rounded off values are the real zeros of the function.

Key concepts

Graphing Utility

A graphing utility is a powerful tool for visualizing mathematical functions. And it's not just limited to your basic everyday calculator.
Think of graphing utilities as smart tools that allow you to see how functions behave. They can plot equations and show x and y-intercepts, turning equations into pictures, which makes them easy to understand.

• **Plot Function**: You can enter a function, like a polynomial, and let the utility plot it on a graph.
• **Adjust Window**: Choose an area to "see" by adjusting the window size. This helps to focus on interesting parts of the graph, like where it crosses the x-axis.
• **Interactive Features**: Many graphing utilities offer interactive features such as zoom and trace. These tools help you dive deeper into the graph to gather more information.

By using a graphing utility, you gain insight into finding zeros, understanding roots, and other key function behaviors. This is invaluable in both learning and solving advanced math problems.

Polynomial Function

Polynomial functions are expressions made up of variables raised to different powers and their coefficients. They are like a combination of terms, such as in the function given: \[-x^3 + 2x^2 + 4x + 5\].

These functions can have different characteristics depending on their terms:

• **Degree of the Polynomial**: The highest power of the variable. Here, it's 3 (cubic), dictating the overall shape of the graph.
• **Coefficients**: Numbers multiplied by the variables, affecting the steepness and direction of the graph.
• **Intercepts and Roots**: Points where the graph intersects the x-axis. These are the zeros or roots of the polynomial.

Polynomials can have multiple zeros, and these are the x-values where a function equals zero. Graphing utilities help you spot these zeros, find intercepts, and understand turning points in polynomial functions. For students, mastering polynomial functions lays a foundation in math that is essential for tackling more complex topics like calculus.

Approximation Methods

When you're working with functions, sometimes it's crucial to find values that aren't exact but rather very close. This is where approximation methods come into play.

Specifically, in this problem, approximation is used to find the real zeros of the polynomial function with precision up to the nearest thousandth.

• **Zooming In**: Using the zoom function allows for a closer look at where the graph intersects the x-axis.
• **Tracing the Graph**: The trace feature is excellent for pinpointing the coordinates close to these intersections. It helps track these near the x-axis precisely.
• **Rounding Off**: Once you identify the approximate zeros, you round them off to the nearest thousandth place. Such practice is helpful when an exact calculation isn't possible.

By mastering approximation methods using graphing utilities, students gain the skill of approaching real-world problems where precision is crucial but not readily exact. This art of approximation becomes a pivotal tool especially in fields of engineering, physics, and computational sciences.