Analyzing The Polynomial Function F(x) = X^4 - 4x^3 + 4x^2 End Behavior And Roots

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In this comprehensive exploration, we delve into the intricacies of the polynomial function f(x) = x⁴ - 4x³ + 4x². Our primary objective is to dissect its behavior, identify its roots, and ultimately, sketch its graph. We will leverage a range of analytical tools, including the Leading Coefficient Test, factoring techniques, and graphical analysis, to gain a thorough understanding of this function. Let's embark on this mathematical journey!

a. Determining End Behavior Using the Leading Coefficient Test

The leading coefficient test is a powerful tool that allows us to predict the end behavior of a polynomial function based on its leading term. The leading term is the term with the highest degree, and it dictates the function's behavior as x approaches positive or negative infinity. In our case, the polynomial function is f(x) = x⁴ - 4x³ + 4x². The leading term is x⁴, which has a positive coefficient (1) and an even degree (4).

When a polynomial function has a positive leading coefficient and an even degree, its graph rises to both the left and the right. This means that as x approaches positive infinity (x → ∞), the function's value also approaches positive infinity (f(x) → ∞). Similarly, as x approaches negative infinity (x → -∞), the function's value also approaches positive infinity (f(x) → ∞). Visualize this as the graph opening upwards, like a parabola but potentially with more complex curves in the middle.

To solidify this understanding, let's consider why this behavior occurs. The even degree of the leading term (x⁴) ensures that regardless of whether x is positive or negative, raising it to the fourth power will always result in a positive value. The positive coefficient (1) then scales this positive value, further reinforcing the upward trend as x moves away from zero in either direction. The other terms in the polynomial (-4x³ and 4x²) will influence the function's behavior closer to the origin, but the leading term dominates as x becomes very large or very small. Therefore, the graph of f(x) = x⁴ - 4x³ + 4x² rises to the left and rises to the right. This knowledge provides a crucial starting point for sketching the graph and understanding the function's overall characteristics. It allows us to anticipate the general shape of the curve and to interpret the significance of other features we will identify, such as roots and turning points.

b. Finding the Zeros of the Polynomial

The zeros of a polynomial function, also known as its roots or x-intercepts, are the values of x for which the function equals zero (f(x) = 0). Finding the zeros is a fundamental step in understanding a polynomial's behavior and sketching its graph. To find the zeros of f(x) = x⁴ - 4x³ + 4x², we need to solve the equation x⁴ - 4x³ + 4x² = 0.

The most effective way to solve this equation is by factoring. First, we can factor out the greatest common factor, which is x²:

x²(x² - 4x + 4) = 0

Now, we have a product of two factors that equals zero. This means that either x² = 0 or (x² - 4x + 4) = 0. The first factor, x² = 0, gives us a zero of x = 0. This zero has a multiplicity of 2, because the factor x appears twice (x² = x * x). The second factor, x² - 4x + 4, is a quadratic expression that can be further factored. We recognize it as a perfect square trinomial:

x² - 4x + 4 = (x - 2)²

Setting this factor equal to zero, we get (x - 2)² = 0, which gives us a zero of x = 2. This zero also has a multiplicity of 2, because the factor (x - 2) appears twice. In summary, the zeros of f(x) = x⁴ - 4x³ + 4x² are x = 0 (with multiplicity 2) and x = 2 (with multiplicity 2). The multiplicity of a zero is a critical piece of information, as it tells us how the graph behaves at that point. A zero with an even multiplicity (like 2 in our case) means the graph touches the x-axis at that point but does not cross it. This touching behavior is characteristic of a turning point or a local extremum. On the other hand, a zero with an odd multiplicity would indicate that the graph crosses the x-axis at that point. The zeros we have found, along with their multiplicities, provide us with valuable anchor points for sketching the graph of the polynomial function. We know that the graph will touch the x-axis at x = 0 and x = 2, but it will not cross it. This, combined with the end behavior we determined earlier, gives us a preliminary idea of the shape of the curve.

c. Determining the Maximum Number of Turning Points

Turning points, also known as local extrema, are points on the graph of a polynomial function where the function changes direction, from increasing to decreasing or vice versa. These points represent local maxima (peaks) or local minima (valleys) of the function. The number of turning points a polynomial function can have is related to its degree. The degree of a polynomial is the highest power of x in the function. In our case, f(x) = x⁴ - 4x³ + 4x² has a degree of 4, as the highest power of x is x⁴.

A fundamental property of polynomials states that a polynomial function of degree n can have at most n - 1 turning points. This is a consequence of the fact that the derivative of a polynomial of degree n is a polynomial of degree n - 1, and the turning points correspond to the zeros of the derivative. Applying this rule to our function, which has a degree of 4, the maximum number of turning points is 4 - 1 = 3. It is important to note that this is the maximum number of turning points. The function may have fewer turning points, but it cannot have more. The actual number of turning points depends on the specific coefficients and terms of the polynomial. To find the exact locations of the turning points, we would typically need to use calculus, specifically finding the critical points by setting the derivative of the function equal to zero and solving for x. However, knowing the maximum number of turning points provides us with a valuable constraint when sketching the graph. We know that the graph can change direction at most three times, which helps us to visualize the possible shapes and curves of the function. Combined with the end behavior and the zeros we have already determined, the maximum number of turning points gives us a more complete picture of the function's overall behavior.

d. Sketching the Graph

Sketching the graph of a polynomial function involves bringing together all the information we have gathered so far. This includes the end behavior, the zeros (with their multiplicities), and the maximum number of turning points. Let's recap what we know about f(x) = x⁴ - 4x³ + 4x²:

  • End Behavior: The graph rises to the left and rises to the right (because the leading coefficient is positive and the degree is even).
  • Zeros:
    • x = 0 (multiplicity 2): The graph touches the x-axis at this point but does not cross it.
    • x = 2 (multiplicity 2): The graph touches the x-axis at this point but does not cross it.
  • Maximum Number of Turning Points: 3

With this information, we can start to sketch the graph. First, we plot the zeros on the x-axis: x = 0 and x = 2. Since the graph touches the x-axis at both of these points, we know that these are turning points. Next, we consider the end behavior. The graph rises to the left, so as we move from left to right, the graph will come down from positive infinity towards the x-axis. It touches the x-axis at x = 0, then turns around and goes back up. Since we have another zero at x = 2, the graph must come back down and touch the x-axis at x = 2. Finally, since the graph also rises to the right, it will turn around at x = 2 and continue upwards towards positive infinity. Now, let's think about the turning points. We know there can be at most 3 turning points. We have already identified two turning points at the zeros (x = 0 and x = 2). This means there can be at most one additional turning point between these zeros. This turning point will be a local minimum, as the graph must go down and then back up between the two zeros. To get a more accurate sketch, we could find the y-coordinate of this local minimum. This could be done by estimating the location of the minimum (perhaps around x = 1) and plugging it into the function, or by using calculus to find the exact location. However, for a sketch, we can simply draw a smooth curve that reflects the general behavior we have described. The sketch will show a graph that rises from the left, touches the x-axis at x = 0, goes down to a local minimum, comes back up to touch the x-axis at x = 2, and then rises to the right. This provides a visual representation of the polynomial function's behavior. The sketch is a powerful tool for understanding the function's overall shape and characteristics. It complements the analytical information we have gathered, giving us a complete picture of the polynomial function.

e. Verifying the Graph with Technology

In the age of technology, we have access to a variety of tools that can help us verify our understanding of mathematical concepts. Graphing calculators and online graphing utilities are invaluable resources for checking the accuracy of our sketches and exploring the behavior of functions. To verify the graph of f(x) = x⁴ - 4x³ + 4x², we can use a graphing calculator or an online tool like Desmos or GeoGebra. These tools allow us to input the function and generate a visual representation of its graph. When we graph the function, we should observe the following:

  • The graph rises to the left and rises to the right, confirming our analysis using the Leading Coefficient Test.
  • The graph touches the x-axis at x = 0 and x = 2, but does not cross it. This confirms our findings about the zeros and their multiplicities.
  • The graph has a local minimum between x = 0 and x = 2. This confirms the existence of an additional turning point.

By comparing the graph generated by the technology with our sketch, we can assess the accuracy of our work. If there are discrepancies, we can re-examine our analysis and identify any errors or omissions. For example, we can check if we have correctly identified the zeros, the end behavior, and the number of turning points. Technology can also help us to refine our sketch. We can zoom in on specific regions of the graph to get a more detailed view of the turning points and the behavior near the zeros. We can also use the graphing tool to find the coordinates of the local minimum, providing a more precise understanding of the function's behavior. Furthermore, graphing utilities often have features that allow us to analyze the function in more detail, such as finding the roots, maxima, and minima directly. This can be a valuable way to reinforce our understanding and to check our analytical results. In conclusion, verifying the graph with technology is an essential step in the process of understanding polynomial functions. It allows us to confirm our analysis, refine our sketch, and gain a deeper appreciation for the function's behavior. It bridges the gap between analytical calculations and visual representation, providing a complete and comprehensive understanding of the function.

Conclusion

In this exploration of the polynomial function f(x) = x⁴ - 4x³ + 4x², we have utilized a combination of analytical techniques and graphical tools to gain a comprehensive understanding of its behavior. We have successfully determined the end behavior using the Leading Coefficient Test, found the zeros and their multiplicities through factoring, calculated the maximum number of turning points, sketched the graph, and verified our results with technology. This process demonstrates a systematic approach to analyzing polynomial functions, which can be applied to a wide range of similar problems. By mastering these techniques, we can confidently tackle more complex polynomial functions and appreciate their diverse and fascinating properties.