Graph F(x) = 2x³ - 26x - 24 Behavior Explained

Hey everyone! Today, we're diving deep into the fascinating world of polynomial functions, specifically focusing on how to describe the behavior of a graph. We'll be using the function f(x) = 2x³ - 26x - 24 as our case study. If you've ever wondered how to determine where a graph sits above or below the x-axis, or where it's increasing or decreasing, you're in the right place! Let's break down the process step by step and make sure you're a pro at analyzing cubic functions by the end of this guide.

Understanding Polynomial Functions

Polynomial functions are the building blocks of many mathematical models, and understanding them is crucial for various applications in science, engineering, and even economics. Before we jump into our specific function, let's recap some key concepts. A polynomial function is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. In simpler terms, it's a function like f(x) = axⁿ + bxⁿ⁻¹ + ... + c, where a, b, and c are constants, and n is a non-negative integer. Our function, f(x) = 2x³ - 26x - 24, perfectly fits this description, making it a cubic polynomial function due to the highest power of x being 3. The degree of a polynomial (here, 3) gives us important clues about the graph's shape and behavior, such as the maximum number of turning points it can have (which is one less than the degree, so 2 in our case). Analyzing polynomial functions often involves finding their roots (where the function equals zero), understanding their end behavior (what happens as x approaches positive or negative infinity), and determining intervals of increasing or decreasing behavior.

One of the first things we need to figure out about a polynomial function is its roots, also known as zeros or x-intercepts. These are the points where the graph crosses or touches the x-axis, i.e., where f(x) = 0. Finding roots often involves factoring the polynomial, which can sometimes be tricky, especially for cubic functions. For our function, f(x) = 2x³ - 26x - 24, we can start by factoring out a common factor of 2: f(x) = 2(x³ - 13x - 12). Now, we need to find the roots of the cubic expression x³ - 13x - 12. This can be done using methods like the Rational Root Theorem or synthetic division. The Rational Root Theorem helps us identify potential rational roots (roots that can be expressed as a fraction). By testing factors of the constant term (-12), we might find that x = -3, x = -1, and x = 4 are roots of the cubic. This means that (x + 3), (x + 1), and (x - 4) are factors of the cubic. Factoring completely, we have f(x) = 2(x + 3)(x + 1)(x - 4). These roots are critical points that help us sketch the graph and understand its behavior. Each root corresponds to a point where the graph intersects the x-axis, and the multiplicity of each root (how many times it appears as a factor) tells us something about how the graph behaves at that point. For example, a root with multiplicity 1 (like all our roots here) means the graph crosses the x-axis at that point.

End behavior is another crucial aspect of analyzing polynomial functions. The end behavior describes what happens to the function's values (f(x)) as x approaches positive or negative infinity. For polynomial functions, the end behavior is primarily determined by the leading term (the term with the highest power of x). In our case, the leading term is 2x³. As x becomes very large (approaches positive infinity), 2x³ also becomes very large and positive. This means that the graph of f(x) rises to the right. Conversely, as x becomes very small (approaches negative infinity), 2x³ becomes very large in the negative direction. Therefore, the graph falls to the left. This understanding of end behavior is essential for sketching a general picture of the graph. We know that our graph will come from negative infinity on the left, cross the x-axis at x = -3, x = -1, and x = 4, and then head towards positive infinity on the right. Combining this with our knowledge of the roots, we can start to visualize the overall shape of the curve. The cubic nature of the function tells us that it will have up to two turning points (local maxima or minima), which are the points where the graph changes direction.

Determining Intervals Above and Below the x-axis

Now, let's get to the core of our task: figuring out where the graph of f(x) = 2x³ - 26x - 24 lies above or below the x-axis. This is also known as determining the intervals where f(x) is positive or negative. The roots we found earlier (x = -3, x = -1, and x = 4) divide the x-axis into intervals. Within each interval, the function will either be entirely above the x-axis (f(x) > 0) or entirely below the x-axis (f(x) < 0). To determine the sign of f(x) in each interval, we can use a sign chart or test points. A sign chart is a table that helps us organize our thoughts and visualize the sign of f(x) in each interval. We list the intervals created by the roots and then choose a test point within each interval. By evaluating f(x) at each test point, we can determine whether the function is positive or negative in that interval. For example, let's consider the interval (-∞, -3). We can choose a test point like x = -4. Plugging this into our factored function, f(x) = 2(x + 3)(x + 1)(x - 4), we get f(-4) = 2(-4 + 3)(-4 + 1)(-4 - 4) = 2(-1)(-3)(-8) = -48, which is negative. This tells us that the graph is below the x-axis in the interval (-∞, -3). We repeat this process for the other intervals: (-3, -1), (-1, 4), and (4, ∞). In the interval (-3, -1), we might choose x = -2 as a test point. Calculating f(-2) gives us a positive value, so the graph is above the x-axis in this interval. Similarly, we can find that the graph is below the x-axis in (-1, 4) and above the x-axis in (4, ∞). These findings give us a clear picture of how the graph behaves relative to the x-axis.

Constructing the Table

Now that we've done the hard work, we can construct the table that describes the behavior of the graph of f(x) = 2x³ - 26x - 24. Our table will have two columns: one for the intervals defined by the roots and one for the relation of the graph to the x-axis in each interval. We found that the roots are x = -3, x = -1, and x = 4, so our intervals are (-∞, -3), (-3, -1), (-1, 4), and (4, ∞). Using our test points and sign analysis, we determined that:

• In the interval (-∞, -3), the graph is below the x-axis.
• In the interval (-3, -1), the graph is above the x-axis.
• In the interval (-1, 4), the graph is below the x-axis.
• In the interval (4, ∞), the graph is above the x-axis.

Here’s what the table looks like:

This table neatly summarizes the behavior of the graph concerning the x-axis. It shows us where the function is positive (above the x-axis) and where it is negative (below the x-axis). This information is incredibly valuable for sketching the graph and understanding the function's properties.

Putting It All Together

To recap, we've taken a deep dive into analyzing the behavior of the graph of f(x) = 2x³ - 26x - 24. We started by understanding the basics of polynomial functions, including their roots and end behavior. We then focused on determining the intervals where the graph is above or below the x-axis using test points and sign analysis. Finally, we compiled our findings into a table that clearly describes the graph's behavior. This process can be applied to any polynomial function, making it a powerful tool for understanding and visualizing mathematical relationships. Remember, the key is to break the problem down into smaller, manageable steps: find the roots, analyze the end behavior, determine the sign in each interval, and then compile your results. With practice, you'll become a pro at decoding the behavior of graphs! Keep exploring, guys, and happy graphing!