Problem 45 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function and to approximate any relative minimum or relative maximum values of the function. $$f(x)=3 x^{2}-6 x+1$$

Step-by-Step Solution

Answer

The function has a relative minimum at about x=1, as shown by the graph.

Step 1: Graph the Function

Enter the function \(f(x)=3x^{2}-6x+1\) into the graphing utility and draw the graph. Observe the curve of the graph.

Step 2: Identify Relative Minima/Maxima by Observing the Curve

Look at the graph and identify any relative minima or maxima. These are the lowest or highest points, respectively, on any particular section of the graph, not considering the whole domain of the function. A relative minimum is where the graph changes from decreasing to increasing, and a relative maximum is where the graph changes from increasing to decreasing.

Step 3: Approximate Relative Minima/Maxima

Use your graphing utility to approximate the x-values at which the minimum or maximum occurs. For more precision, you could use calculus methods, but approximation from the graph suffices for this exercise.

Key Concepts

Relative Minimum and Maximum

Understanding the points of a quadratic function where it reaches its lowest and highest values on a particular section can be crucial for grasping the nature of these functions. These points are known as the relative minimum and maximum.

Visualizing this concept, imagine walking along a hilly path—the point where you are at the bottom of a valley would be a relative minimum because it's the lowest point in that area. Similarly, when you reach the top of a hill before descending, you're at a relative maximum. In the graph of a quadratic function, relative minimums and maximums appear where the curve changes direction, which is also where the slope of the function is zero.

To identify these points on the graph of the quadratic function like \(f(x)=3x^{2}-6x+1\), use a graphing utility or plot points manually to see where the graph naturally rises to a peak or dips to a trough. Being able to approximate the coordinates of these points will give you insights into the function's behavior, especially in applied contexts like physics or economics.

Graphing Utility

A graphing utility is an invaluable tool that provides a visual representation of functions, aiding in the analysis of their characteristics, such as intercepts, slopes, relative minima, and maxima.

Modern graphing utilities can range from physical calculators, like the TI-84, to software applications or online platforms that can plot complex equations with ease. These tools allow for dynamic interaction with the graph—you can zoom in for a detailed view, adjust the scale for better perspective, and even animate the changes to a function as you tweak its parameters.

When you input the function \(f(x)=3x^{2}-6x+1\) into a graphing utility, it will produce a parabola, and you can then use features like 'Trace' or 'Analyze Graph' to pinpoint relative minimums or maximums with precision. Grasping how to use these utilities enhances your understanding of quadratic functions and makes the complex simple.

Quadratic Functions

Quadratic functions are a type of polynomial function with a degree of two. The general form of a quadratic function is \(ax^{2}+bx+c\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is nonzero. These functions graph to a U-shaped curve known as a parabola.

The parabola can open upwards or downwards, dependent on the sign of the \(a\) coefficient—if positive, it opens upward, and if negative, it opens downward. The function \(f(x)=3x^{2}-6x+1\) will produce an upwards opening parabola, indicating it will have a relative minimum but no relative maximum.

Roots, or zeroes, of quadratic functions are found where the graph intersects the x-axis. The vertex, the highest or lowest point on the graph depending on the direction the parabola opens, gives the function’s maximum or minimum value. Knowing the properties of quadratic functions not only helps in sketching their graphs but also in understanding real-world situations where they apply, such as projectile motion or market economics.