Problem 30 - 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. $$ f(x)=x^{5}+4 x^{4}-x^{3}-9 x^{2}+3 $$

Step-by-Step Solution

Answer

Graph the function, find zeros between integers by crossing points, and estimate peaks and valleys from the graph.

Step 1: Create a Table of Values

To graph the function \( f(x) = x^5 + 4x^4 - x^3 - 9x^2 + 3 \), choose several values of \( x \) and compute the corresponding \( f(x) \). For example, if you choose \( x = -3, -2, -1, 0, 1, 2, 3 \), calculate \( f(x) \) for each. This will help visualize the function's behavior.

Step 2: Graph the Function

Using the table of values from Step 1, plot each point \((x, f(x))\) on a Cartesian plane. Draw a smooth curve through these points to represent the function.

Step 3: Identify Consecutive Integer Values with Real Zeros

Look at your graph from Step 2 to identify where the function crosses the x-axis. Identify the intervals between consecutive integer values of \( x \) where these crossings occur. These intervals indicate the locations of real zeros of the function.

Step 4: Estimate Relative Maxima and Minima

Using the graph, determine the regions where the function changes from increasing to decreasing (relative maxima) and from decreasing to increasing (relative minima). Estimate the \( x \)-coordinates of these points, noting where the peaks and valleys occur.

Key Concepts

Graphing functions

Graphing a function is a powerful way to understand its behavior. To begin, you'll create a table of values. Choose a series of x-values from your function's domain — these are numbers you'll substitute into your function to find the corresponding y-values. For example, graphing the polynomial function \( f(x) = x^5 + 4x^4 - x^3 - 9x^2 + 3 \) involves calculating points like \( (-3, f(-3)) \), \( (0, f(0)) \), \( (3, f(3)) \), etc.

Once you have a list of x, y pairs, plot these points onto a Cartesian coordinate system. After plotting, you'll connect the dots smoothly, taking care to represent any curves or bends in the graded manner observed in polynomial graphs. A polynomial's degree and leading coefficients heavily influence its shape. Ensure the graph reflects the nature of the polynomial, which might mean some trial and error to place points accurately.

These visual tools illustrate shifts, trends, and significant changes in the function, aiding further investigation of its characteristics.

Real zeros

Real zeros of a polynomial function occur where the function crosses or touches the x-axis. These points are x-values where \( f(x) = 0 \). Identifying them involves examining the graph from the previous step. Seek out intersections with the x-axis, as these indicate real zeros.

When locating them, you might not see exact zeros but rather intervals between consecutive integers where zeros lie. For instance, if the curve crosses the x-axis between \( x = 1 \) and \( x = 2 \), this interval contains a real zero. In practice, tools like graphing software or finding exact values via algebraic methods offer precision.

The visual identification on a graphed curve allows you to approximate these intervals, forming a crucial step in understanding the function's root behavior. Different methods can then refine these approximations as needed.

Relative maxima

Relative maxima in a function are the 'peaks.' They're where a function changes direction from increasing to decreasing within a particular interval of its domain. On the graph, these peaks represent local high points, not necessarily the highest point on the entire graph, but the highest in their immediate vicinity.

To locate these, observe where the curve transitions from going upward to moving downward. This can sometimes be subtle, especially if the peak forms a rounded or stretched out bulge. After identifying potential regions, estimate the x-coordinate of these peaks. The exact y-value indicates the maximum height relative to surrounding points.

Understanding where these relative maxima occur can reveal much about the general dynamics and symmetries in a function's behavior.

Relative minima

Relative minima are the 'valleys' in a function's graph. These occur where the function transitions from decreasing to increasing, creating a local low point. Just like relative maxima, relative minima aren't the lowest point over the entire graph, only within their localized region.

To identify them, look for segments where the graph curves upward after descending. These are typically more visible than relative maxima due to the natural dip these curves take. Estimate the corresponding x-values and evaluate the y-values to understand how low the curve dips, relative to neighboring points.

By pinpointing where relative minima exist, you gain insight into the function's periodicity and potential symmetry, enhancing both analysis and prediction regarding function behavior.