Problem 28 - Algebra 2

Question

For Exerises \(26-31\) , complete each of the following. a. Graph each funnction by making a table of values. b. Determine the consecutive integer values of \(x\) between which each real zero is located. C. Estimate the \(x\) -coordinates at which the relative maxima and relative minima occur. $$ f(x)=x^{4}-9 x^{3}+25 x^{2}-24 x+6 $$

Step-by-Step Solution

Answer

Graph the function, estimate zeros between integers, maxima/minima around curve peaks/troughs.

Step 1: Create a Table of Values

To graph the function, start by selecting integer values for \(x\) and calculate the corresponding \(f(x)\). For instance, choose \(x = -2, -1, 0, 1, 2, 3, 4, 5,\) and \(6\). Calculate each \(f(x)\) value using the function \(f(x) = x^4 - 9x^3 + 25x^2 - 24x + 6\). For example: \(f(0) = 6\), \(f(1) = -1\), and \(f(2) = -2\). Repeat this process for other selected \(x\) values.

Step 2: Plot the Points and Graph the Function

Using the table of values from Step 1, plot each point \((x,f(x))\) on a coordinate plane. Draw a smooth curve through these points to graph the function. Observe how the function behaves as \(x\) increases or decreases.

Step 3: Determine Consecutive Integers for Real Zeros

By looking at the graph, find the intervals where the function crosses the \(x\)-axis. These crossings indicate the approximate real zeros of the function. Check where changes from positive to negative \(f(x)\) values occur between consecutive integer \(x\) values. For instance, if \(f(1) < 0\) and \(f(2) > 0\), then a zero occurs between \(x = 1\) and \(x = 2\). Repeat this to find all intervals.

Step 4: Estimate Relative Maxima and Minima

Look for peaks (highest points) and troughs (lowest points) in the graph to estimate the relative maxima and minima. These points occur where the slope of the graph changes from positive to negative (maxima) or negative to positive (minima). Identify approximate \(x\)-coordinates for these turning points by observing where these changes happen visually on the graph.

Key Concepts

Zeros of a Function

To find the zeros of a polynomial function, we're basically looking for the points where the function crosses the x-axis. These points are important because they indicate where the output, or value of the function, is zero.

Finding zeros involves identifying intervals on the graph where the function changes sign, such as going from positive to negative or vice versa. For instance, if we observe that for two consecutive integer values of \(x\), one produced a positive \(f(x)\) and the other produced a negative \(f(x)\), then there must be a zero between these integers. This method helps in approximating the real zeros without solving the equation algebraically.

By carefully analyzing the table of values and graph, one can determine the approximate location of these zeros efficiently. This is a practical way when a function is complex and cannot be easily factored.

Relative Maxima and Minima

The relative maxima and minima of a function refer to the peaks and troughs on a graph. A relative maximum is a point where the function changes from increasing to decreasing, which means the slope of the tangent is zero and flips from positive to negative. Conversely, a relative minimum occurs where the function changes from decreasing to increasing, with the slope changing from negative to positive.

When graphing polynomial functions, identifying these points involves looking for where the curve turns to reach a peak or a valley. This is done visually by examining the graph for points where the curve is highest (maxima) or lowest (minima) relative to their surroundings.

These points are crucial in understanding the behavior of the polynomial, indicating where the maximum or minimum outputs occur, thus providing insights into the function's overall shape.

Table of Values

Creating a table of values is the first step in graphing a polynomial function. This involves choosing several \(x\)-values and computing the corresponding \(f(x)\) values using the given function. By doing so, you gain a better understanding of how the function behaves over a range of \(x\)-values.

To build this table, select a series of integers that are easily computable, such as \(x = -2, -1, 0, 1, 2, 3, 4, 5,\) and \(6\). Then, substitute each \(x\)-value into the function to find the corresponding \(f(x)\). For example, if you plug in \(x = 0\) into \(f(x) = x^4 - 9x^3 + 25x^2 - 24x + 6\), the result is \(f(0) = 6\).

This table helps in plotting the graph accurately, as it provides specific points through which the curve must pass, offering a guide to drawing the polynomial's shape.

X-axis Intersections

When graphing polynomial functions, the x-axis intersections are particularly significant. These intersections are the points where the graph touches or crosses the x-axis, and they are also the zeros of the function. Each intersection represents a solution to the equation \(f(x) = 0\).

To find these, you can use the table of values and observe where \(f(x)\) changes from positive to negative or vice versa between two consecutive x-values. This indicative change is a sign that the graph has passed through the x-axis.

Understanding where these intersections occur not only helps in sketching the function but also in understanding its roots. The x-axis intersections are basically the graphical representation of the function's solutions, providing insights into its real-world applications.