Problem 53 - Algebra 2

Question

Graph each function by making a table of values. $$ f(x)=x^{4}-6 x^{3}+10 x^{2}-x-3 $$

Step-by-Step Solution

Answer

Choose x values, calculate f(x), plot points, and draw the curve.

Step 1: Choose Input Values for x

To graph the function, we first need to select a set of input values for \(x\). A good initial range might be from \(-3\) to \(3\), inclusive. This range ensures we capture the behavior of the polynomial on both sides of the critical points.

Step 2: Calculate Corresponding f(x) Values

Next, for each chosen value of \(x\), we calculate \(f(x)\) using the function \(f(x) = x^{4} - 6x^{3} + 10x^{2} - x - 3\). - For \(x = -3\), \(f(-3) = (-3)^4 - 6(-3)^3 + 10(-3)^2 - (-3) - 3\).- For \(x = -2\), \(f(-2) = (-2)^4 - 6(-2)^3 + 10(-2)^2 - (-2) - 3\).- Continue this method for \(x = -1, 0, 1, 2, 3\).

Step 3: Create the Table of Values

Organize the \(x\) values and their corresponding \(f(x)\) values into a table format:- For example, after calculation: \[-3, -2, -1, 0, 1, 2, 3\] \[ -81, -35, -8, -3, 1, -5, 9\] This table assists in plotting the function on the graph.

Step 4: Plot the Points on a Graph

Using the table of values, plot each point on a coordinate plane. Each point is of the form \((x, f(x))\). For instance, plot the point \((-3, -81)\), \((-2, -35)\), and so on. Connect the points with a smooth curve to represent the polynomial function.

Step 5: Analyze the Graph

Once plotted, the graph of the polynomial will show critical features such as roots, turning points, and end behavior. By examining the curve, observe how it behaves as \(x\) moves from left to right across the chosen range.

Key Concepts

Graphing Polynomials

Graphing polynomials involves creating a visual representation of a polynomial function on a coordinate plane. It begins with choosing a range of input values for \(x\) and using these to calculate the corresponding \(f(x)\) values. These calculated values help plot points on the graph that represent the function. For the function \(f(x) = x^4 - 6x^3 + 10x^2 - x - 3\), we might choose \(x\) values from \(-3\) to \(3\) to capture the function's behavior around critical regions.

To graph efficiently:

• Choose a balanced range of \(x\) values to cover potential peaks and valleys.
• Calculate \(f(x)\) for each chosen \(x\).
• Plot the results as points \((x, f(x))\) on a graph.
• Connect these points smoothly. The curve represents the function.

This graph can help visualize the behavior and characteristic changes in the polynomial function.

Table of Values

Creating a table of values provides a structured way to calculate and organize inputs and corresponding output values for a polynomial function. This aids in graphing as it visually lays out which points to plot.

For our polynomial \(f(x) = x^4 - 6x^3 + 10x^2 - x - 3\), begin by calculating \(f(x)\) for each \(x\) within your chosen range, such as from \(-3\) to \(3\).

For instance:

• \(x = -3\), yields \(f(-3) = -81\)
• \(x = 0\), yields \(f(0) = -3\)
• \(x = 3\), yields \(f(3) = 9\)

Organizing them helps maintain clarity and ensures points can be plotted without error. Such a table forms the skeleton of the graph, dictating where the function's graphical path will lie on the coordinate plane.

Behavior of Polynomial Functions

The behavior of a polynomial function describes how it behaves across different \(x\) values. Key features include how the polynomial rises and falls, the direction of its ends, and its symmetry or lack thereof. These characteristics are revealed through graphing and interpreting tables of values.

**Key Aspects:**

• As \(x\rightarrow +\infty\) or \(-\infty\), observe the function's end behavior; it signifies how the curve approaches infinity or negative infinity.
• Identify peaks, troughs, and plateaus, which indicate turning points and root approximations.
• Consider symmetry; functions might be even or odd based on these characteristics.

Recognizing these aspects helps predict and understand the overall shape and direction of the polynomial on a graph.

Critical Points of Polynomials

In polynomials, critical points are locations on the graph where the function changes direction. These include peaks (maximums), valleys (minimums), and sometimes, a plateau.

Identifying these requires noting where the slope of the graph changes from positive to negative or vice versa, which often indicates the presence of a critical point.

**To Identify Critical Points:**

• Look for points where the derivative is zero, which often correspond to turning points.
• Consider testing values surrounding these points to confirm whether they are maxima or minima.
• Use the original and derivative graphs to confirm these areas.

Recognizing critical points is vital for fully understanding how and why the function behaves in its unique manner across different sections of the graph.