Graphing F(x) = -0.08x(x^2 - 11x + 18): A Step-by-Step Guide

by Pedro Alvarez 61 views

Hey guys! Today, we're diving deep into the fascinating world of functions and graphs, specifically focusing on the function f(x) = -0.08x(x^2 - 11x + 18). Our mission? To figure out which graph perfectly represents this mathematical expression. This might seem daunting at first, but don't worry, we'll break it down step-by-step, making it super easy to understand. We’ll explore key features like x-intercepts, end behavior, and the overall shape of the curve. So, buckle up, grab your thinking caps, and let's get started on this mathematical adventure! By the end of this exploration, you'll not only be able to identify the correct graph but also have a solid understanding of how different components of a function influence its graphical representation.

Unpacking the Function: What Does It Tell Us?

Before we even glance at any graphs, let's get to know our function, f(x) = -0.08x(x^2 - 11x + 18). Understanding the function's structure is like having a roadmap before a journey; it guides us in the right direction. First off, we can see this is a polynomial function. Polynomials are those smooth, continuous curves that don't have any sharp breaks or jumps. This is a crucial piece of information, guys, because it immediately helps us eliminate any graphs that look jagged or discontinuous. The highest power of x in this function will determine its degree, and the degree tells us a lot about the graph's general shape and end behavior. Let's expand the function to clearly see the powers of x:

f(x) = -0.08x(x^2 - 11x + 18) = -0.08x^3 + 0.88x^2 - 1.44x

Ah, there it is! We see that the highest power of x is 3, making this a cubic function. Cubic functions have a distinctive shape, often resembling a stretched-out 'S' curve. They can have up to two turning points (where the graph changes direction) and are known for their interesting end behavior. But what about the end behavior specifically? Well, that's where the leading coefficient comes into play. The leading coefficient is the number in front of the highest power of x, which in our case is -0.08. Since it's negative, the graph will fall (go down) to the right and rise (go up) to the left. Think of it like reading a book – if the leading coefficient is negative, the story ends on a down note (falling to the right). This knowledge is gold because it helps us narrow down our choices significantly. We now know to look for a graph that has the characteristic 'S' shape of a cubic function and exhibits this specific end behavior. Next, we’ll dig into finding the x-intercepts, which are another crucial piece of the puzzle. Understanding these fundamental aspects sets the stage for accurately matching the function to its graph.

Finding the X-Intercepts: Where the Graph Crosses the Axis

Now, let's talk x-intercepts. These are the points where the graph of the function crosses the x-axis. In simpler terms, they're the values of x for which f(x) equals zero. Finding these intercepts is super important because they give us specific points that the graph must pass through. To find the x-intercepts, we need to solve the equation f(x) = 0. So, let's set our function to zero and see what we get:

0 = -0.08x(x^2 - 11x + 18)

This looks manageable! We've got a product of terms equaling zero, which means that at least one of those terms must be zero. The first term, -0.08x, gives us one intercept right away:

-0.08x = 0 => x = 0

So, x = 0 is one of our x-intercepts. This means the graph passes through the origin (0, 0). Cool! Now, let's tackle the quadratic part, (x^2 - 11x + 18). To find the roots (the values of x that make it zero), we can either try to factor it or use the quadratic formula. Factoring looks promising here. We need two numbers that multiply to 18 and add up to -11. Those numbers are -2 and -9. So, we can factor the quadratic as follows:

(x^2 - 11x + 18) = (x - 2)(x - 9)

Awesome! Now, we set each factor to zero:

(x - 2) = 0 => x = 2

(x - 9) = 0 => x = 9

Alright! We've found our x-intercepts: x = 0, x = 2, and x = 9. This is fantastic information! We now know that the graph must cross the x-axis at these three points. When we look at the potential graphs, we can immediately eliminate any that don't cross the x-axis at these exact locations. Armed with the x-intercepts and our understanding of the function's end behavior, we're getting closer to pinpointing the correct graph. In the next section, we’ll consider the behavior of the graph between the intercepts and how the leading coefficient influences the overall shape.

The Shape Between Intercepts: Understanding the Curve

We've nailed down the x-intercepts (0, 2, and 9) and the end behavior (falling to the right, rising to the left). Now, let's focus on what happens to the graph between these intercepts. This is where we can really see the cubic nature of our function, f(x) = -0.08x(x^2 - 11x + 18), come to life. Remember, guys, that cubic functions can have up to two turning points, where the graph changes from increasing to decreasing or vice versa. These turning points are crucial in defining the curve's shape. To understand the behavior between the intercepts, it's helpful to think about the sign of f(x) in each interval. Let's consider the intervals created by our x-intercepts:

  1. x < 0: In this interval, x is negative. Looking at the factored form, -0.08x is positive, (x - 2) is negative, and (x - 9) is negative. So, the overall product is positive (positive * negative * negative = positive). This means the graph is above the x-axis to the left of x = 0.
  2. 0 < x < 2: Here, x is positive, so -0.08x is negative. (x - 2) is negative, and (x - 9) is negative. The product is negative (negative * negative * negative = negative). The graph is below the x-axis between x = 0 and x = 2.
  3. 2 < x < 9: In this interval, x is positive, -0.08x is negative. (x - 2) is positive, and (x - 9) is negative. The product is positive (negative * positive * negative = positive). The graph is above the x-axis between x = 2 and x = 9.
  4. x > 9: Here, x is positive, -0.08x is negative. (x - 2) is positive, and (x - 9) is positive. The product is negative (negative * positive * positive = negative). The graph is below the x-axis to the right of x = 9.

This sign analysis gives us a clear picture of the graph's behavior. It starts positive (above the x-axis), crosses at x = 0, goes negative (below the x-axis), crosses at x = 2, goes positive again, crosses at x = 9, and finally goes negative again. This information, combined with the end behavior, allows us to visualize the general shape of the graph. We're looking for an 'S' shaped curve that rises from the left, crosses the x-axis at 0, dips below, rises again to cross at 2, peaks above the x-axis, falls to cross at 9, and then continues downward. By now, we have a pretty solid idea of what the graph should look like. All that's left is to compare our mental image with the given options and choose the best match.

Putting It All Together: Choosing the Correct Graph

Alright, guys, we've done the groundwork! We know our function, f(x) = -0.08x(x^2 - 11x + 18), inside and out. We've figured out the x-intercepts (0, 2, and 9), understood the end behavior (falls to the right, rises to the left), and analyzed the curve's shape between the intercepts. Now comes the exciting part: matching all this knowledge to the correct graph. When you're presented with multiple graph options, it's like being a detective with a set of clues. Each piece of information we've gathered is a clue that helps us eliminate the wrong suspects and pinpoint the right one. Here’s a quick recap of our clues:

  • Cubic Function: Look for a general 'S' shape.
  • X-Intercepts: The graph must cross the x-axis at x = 0, x = 2, and x = 9.
  • End Behavior: The graph rises to the left and falls to the right (due to the negative leading coefficient).
  • Behavior Between Intercepts: Above the x-axis to the left of 0, below between 0 and 2, above between 2 and 9, and below to the right of 9.

Armed with these clues, you can systematically go through each graph option. Start by checking the x-intercepts. Does the graph cross the x-axis at 0, 2, and 9? If not, eliminate it immediately. Next, verify the end behavior. Does the graph rise to the left and fall to the right? If not, scratch it off the list. Finally, examine the curve's behavior between the intercepts. Does it match our sign analysis? If a graph ticks all these boxes, congratulations! You've likely found the correct representation of the function. If multiple graphs seem to fit the criteria, look for subtle differences in the curve's shape or the steepness of the slopes. The more precise your analysis, the more confident you can be in your choice. Remember, the key is to use all the information we've gathered to make an informed decision. By systematically comparing the graphs to our understanding of the function, we can confidently identify the correct match. Great job, guys! We’ve tackled this problem like pros, and hopefully, you’ve gained a deeper appreciation for the connection between functions and their graphical representations.

Final Thoughts: The Power of Understanding Function Behavior

So, there you have it! We've successfully navigated the process of identifying the graph of the function f(x) = -0.08x(x^2 - 11x + 18). This wasn't just about finding the right answer; it was about understanding the why behind the graph's shape. We've seen how factoring helps us find x-intercepts, how the leading coefficient dictates end behavior, and how analyzing intervals reveals the curve's movement above and below the x-axis. These are powerful tools, guys, and they apply to many different types of functions. The key takeaway here is that understanding the fundamental characteristics of a function – its intercepts, degree, leading coefficient, and intervals of positivity and negativity – allows us to predict and interpret its graphical representation. This skill is invaluable not just in mathematics but also in various fields like physics, engineering, and data analysis, where understanding graphical representations is crucial for interpreting data and making informed decisions. Remember, math isn't just about memorizing formulas; it's about developing a deep understanding of concepts and how they connect. By mastering these fundamental principles, you'll be well-equipped to tackle even more complex problems in the future. So, keep exploring, keep questioning, and keep building your mathematical intuition. You've got this!