Analyzing The Graph Of F(x) = X³ - 4x² - 3x + 18

Hey guys! Let's dive into analyzing the function f(x) = x³ - 4x² - 3x + 18. We're going to break down its graph and figure out which statements accurately describe it. This is super important for understanding polynomial functions and how they behave. We'll be focusing on things like roots (where the graph hits the x-axis), double roots (where the graph touches the x-axis and turns around), and how the function behaves as x gets really big or really small. So, buckle up, and let's get started!

Analyzing Solutions and Roots

When we're talking about solutions to the function f(x) = x³ - 4x² - 3x + 18 when y = 0, we're essentially asking: Where does this graph cross the x-axis? These points are also known as the roots or zeros of the function. Finding these roots is a crucial part of understanding the graph's behavior. To determine if there are three unique solutions, we'd typically try to factor the polynomial or use numerical methods to find the roots. Factoring a cubic polynomial can be a bit tricky, but it's a worthwhile endeavor. One method is to try synthetic division with potential rational roots (using the Rational Root Theorem). If we find a root, say x = a, then (x - a) is a factor, and we can divide the polynomial by (x - a) to get a quadratic, which is much easier to solve. If the resulting quadratic has two distinct real roots, then we have three unique solutions. However, if the quadratic has a double root or no real roots, the situation changes. So, let's put on our detective hats and start searching for those roots! To start, we can test some simple integer values. Let's try x = 2: f(2) = (2)³ - 4(2)² - 3(2) + 18 = 8 - 16 - 6 + 18 = 4. So, x = 2 is not a root. Let's try x = 3: f(3) = (3)³ - 4(3)² - 3(3) + 18 = 27 - 36 - 9 + 18 = 0. Bingo! x = 3 is a root. This means (x - 3) is a factor. Now, we can use synthetic division or polynomial long division to divide x³ - 4x² - 3x + 18 by (x - 3). After performing the division, we get x² - x - 6. Now, we need to solve the quadratic equation x² - x - 6 = 0. This can be factored as (x - 3)(x + 2) = 0. So, the roots are x = 3 and x = -2. Notice anything interesting? We have x = 3 as a root twice! This is a big clue about the graph's behavior.

Unpacking Double Roots

The concept of a double root is super important when we're looking at graphs of polynomial functions. A double root, also known as a repeated root or a root with multiplicity 2, occurs when a factor appears twice in the factored form of the polynomial. In our case, we found that f(x) = x³ - 4x² - 3x + 18 has a factor of (x - 3) twice. This means that the graph of the function touches the x-axis at x = 3 but doesn't cross it. Think of it like the graph bouncing off the x-axis at that point. This creates a characteristic “turning point” on the graph. To fully grasp why this happens, let’s think about what happens near the root. When x is slightly less than 3, the factor (x - 3) is negative. When x is slightly greater than 3, the factor (x - 3) is positive. However, because (x - 3) appears twice (as (x - 3)²), the overall sign of the term remains the same on both sides of x = 3. This prevents the graph from crossing the x-axis. On the other hand, at the root x = -2, the factor (x + 2) appears only once. As x moves from less than -2 to greater than -2, the factor (x + 2) changes sign, causing the function to change sign and cross the x-axis. Visualizing this really helps! Imagine the graph approaching the x-axis at x = 3. Instead of slicing through it, it gently touches the axis and then turns back in the direction it came from. This behavior is a hallmark of double roots and gives us valuable information about the shape of the graph. Recognizing double roots is a powerful tool for sketching polynomial graphs and understanding their behavior. It tells us not only where the graph intersects the x-axis but also how it interacts with it.

End Behavior: x Approaching Infinity

Understanding the end behavior of a function is like having a sneak peek at what the graph does way out on the fringes. Specifically, we want to know what happens to f(x) as x gets incredibly large (approaches positive infinity) and incredibly small (approaches negative infinity). For polynomial functions, the term with the highest power (the leading term) dictates the end behavior. In our function, f(x) = x³ - 4x² - 3x + 18, the leading term is x³. This tells us a lot. The degree of the polynomial (the highest power of x) is 3, which is odd. This means the end behavior will be different on the left and right sides of the graph. The leading coefficient (the number in front of the x³) is 1, which is positive. This tells us the direction the graph will go as x approaches positive infinity. When x is a large positive number, x³ will be an even larger positive number. So, as x approaches positive infinity, f(x) also approaches positive infinity. Think of it like this: the graph shoots up to the top-right corner of the coordinate plane. Now, let's consider what happens as x approaches negative infinity. When x is a large negative number, x³ will be a large negative number (because a negative number raised to an odd power is negative). So, as x approaches negative infinity, f(x) also approaches negative infinity. This means the graph plunges down to the bottom-left corner of the coordinate plane. Putting it all together, we know that the graph of f(x) = x³ - 4x² - 3x + 18 starts low on the left, does some interesting things in the middle (like hitting the x-axis at roots and bouncing off at double roots), and then shoots high on the right. This overall shape is characteristic of cubic functions with a positive leading coefficient. Understanding end behavior is a fantastic way to get a general sense of a polynomial's graph before you even start plotting points or using a calculator. It helps you anticipate the overall trend and gives you a framework for interpreting the rest of the graph's features.

Conclusion: Putting It All Together

Okay, guys, we've really dug deep into the function f(x) = x³ - 4x² - 3x + 18, and we've uncovered some key insights about its graph. We figured out its roots, understood the concept of a double root, and analyzed its end behavior. Let's recap what we've learned and see which statements correctly describe the graph:

• Roots: We found that the function has roots at x = 3 (a double root) and x = -2. This means the graph touches the x-axis at x = 3 and crosses it at x = -2.
• Double Root: The presence of a double root at x = 3 is super important. It tells us the graph bounces off the x-axis at that point, creating a turning point.
• End Behavior: As x approaches positive infinity, f(x) also approaches positive infinity. As x approaches negative infinity, f(x) also approaches negative infinity. This gives us the overall direction of the graph – starting low on the left and going high on the right.

With this knowledge, we can confidently evaluate the initial statements and select the ones that accurately describe the graph of the function. Remember, understanding these concepts—roots, double roots, and end behavior—is fundamental to mastering polynomial functions. Keep practicing, keep exploring, and you'll become a graph-analyzing pro in no time!