Find The Graph Of F(x) = -0.08x(x^2 - 11x + 18) Math Problem Solved

by ADMIN 68 views
Iklan Headers

Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of polynomial functions and their graphical representations. Specifically, we're going to dissect the function f(x) = -0.08x(x^2 - 11x + 18) and explore how to identify its graph. This might seem daunting at first, but trust me, with a systematic approach, you'll be able to conquer any polynomial graph! So, let's put on our detective hats and unravel the mysteries of this function.

Understanding the Function's Anatomy

Before we even think about sketching a graph, it's crucial to understand the building blocks of our function. Our function, f(x) = -0.08x(x^2 - 11x + 18), is a polynomial function. Polynomials are expressions consisting of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. They are the workhorses of mathematical modeling, appearing in countless applications across science, engineering, and economics.

The first thing to notice is the degree of the polynomial. The degree is the highest power of the variable in the expression. To find the degree, we need to expand the function. Let's do that:

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

Now it's clear: the highest power of x is 3, so this is a cubic function (a polynomial of degree 3). The degree tells us a lot about the general shape of the graph. Cubic functions typically have a characteristic "S" shape, but the specifics can vary depending on the coefficients.

Next, let's factor the quadratic expression inside the parentheses:

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

So, our function can be rewritten as:

f(x) = -0.08x(x - 2)(x - 9)

This factored form is incredibly valuable because it reveals the zeros (or roots) of the function. The zeros are the values of x for which f(x) = 0. By setting each factor to zero, we find the zeros:

  • -0. 08x = 0 => x = 0
  • x - 2 = 0 => x = 2
  • x - 9 = 0 => x = 9

These zeros are the points where the graph intersects the x-axis. Knowing the zeros gives us a crucial framework for sketching the graph.

Finally, let's consider the leading coefficient, which is the coefficient of the term with the highest power of x. In our expanded form, the leading coefficient is -0.08. The sign of the leading coefficient tells us about the end behavior of the graph:

  • If the leading coefficient is positive, the graph rises to the right (as x approaches positive infinity) and falls to the left (as x approaches negative infinity).
  • If the leading coefficient is negative, the graph falls to the right and rises to the left.

In our case, the leading coefficient is negative, so the graph will fall to the right and rise to the left. This is another vital clue in piecing together the graph.

Key Features for Graph Identification

Okay, guys, now that we've dissected the function, let's summarize the key features we'll use to identify its graph:

  1. Degree: The function is cubic (degree 3), indicating a general "S" shape.
  2. Zeros: The function has zeros at x = 0, x = 2, and x = 9. These are the x-intercepts of the graph.
  3. Leading Coefficient: The leading coefficient is negative (-0.08), so the graph falls to the right and rises to the left.

With these features in mind, we can eliminate many potential graphs. For instance, any graph that isn't roughly "S" shaped or doesn't cross the x-axis at 0, 2, and 9 can be immediately ruled out. Similarly, a graph that rises to the right and falls to the left is not a match for our function.

The Art of Sketching (A Mental Picture)

Before looking at any actual graphs, it's helpful to try to visualize what the graph of our function might look like. We know it's a cubic function with zeros at 0, 2, and 9, and it falls to the right and rises to the left. We can imagine the graph:

  1. Starting high on the left side (as x approaches negative infinity).
  2. Crossing the x-axis at x = 0.
  3. Turning around and crossing the x-axis again at x = 2.
  4. Turning again and crossing the x-axis for the final time at x = 9.
  5. Continuing downward as x approaches positive infinity.

This mental picture gives us a framework for evaluating potential graphs. We're looking for a graph that exhibits this general behavior.

Analyzing Potential Graphs

Now, let's imagine we're presented with a set of graphs and need to identify the one that matches our function. We'll use our key features as a checklist:

  1. Shape: Does the graph have a general "S" shape, characteristic of cubic functions? If not, it's not the graph of our function.
  2. X-intercepts: Does the graph cross the x-axis at x = 0, x = 2, and x = 9? If not, it's not the graph of our function.
  3. End Behavior: Does the graph fall to the right and rise to the left? If not, it's not the graph of our function.

By systematically checking these features, we can narrow down the possibilities and identify the correct graph. It's like a process of elimination, where we rule out graphs that don't fit the criteria until we're left with the one that does.

To make things even easier, we can also consider a few additional points on the graph. For example, we could calculate the value of f(x) at x = 1, x = 3, or x = 10. These points would give us a more precise idea of the graph's shape and position.

Common Pitfalls to Avoid

Identifying graphs can be tricky, and there are a few common pitfalls to watch out for:

  • Misinterpreting the Shape: While cubic functions generally have an "S" shape, the exact shape can vary. Don't get fixated on a perfect "S"; focus on the overall trend and the key features.
  • Ignoring the End Behavior: The end behavior (whether the graph rises or falls to the left and right) is a crucial clue. Make sure the graph matches the sign of the leading coefficient.
  • Overlooking the Zeros: The zeros are the most important points on the graph. Double-check that the graph crosses the x-axis at the correct locations.
  • Rushing to a Conclusion: Take your time and carefully consider all the features before making a decision. It's better to be thorough than to make a hasty mistake.

Example Scenario

Let's walk through a hypothetical scenario to solidify our understanding. Imagine we have four graphs labeled A, B, C, and D. We need to determine which one represents our function, f(x) = -0.08x(x^2 - 11x + 18).

  1. Graph A: This graph is a straight line. Straight lines represent linear functions (degree 1), not cubic functions (degree 3). So, Graph A is not the answer.
  2. Graph B: This graph has a general "S" shape, which is promising. However, it crosses the x-axis at -1, 3, and 8. These are not the zeros of our function (0, 2, and 9). So, Graph B is not the answer.
  3. Graph C: This graph also has a general "S" shape and crosses the x-axis at 0, 2, and 9. This is a good sign! It also falls to the right and rises to the left, matching the negative leading coefficient. Graph C looks like a strong contender.
  4. Graph D: This graph has an "S" shape and crosses the x-axis at 0, 2, and 9. However, it rises to the right and falls to the left. This contradicts the negative leading coefficient. So, Graph D is not the answer.

Based on our analysis, Graph C is the most likely candidate. It exhibits the correct shape, zeros, and end behavior. To be absolutely sure, we could check a few additional points, but in this case, Graph C is almost certainly the correct answer.

Level Up Your Graphing Skills

Identifying graphs of functions is a fundamental skill in mathematics. It connects algebraic expressions to visual representations, allowing us to understand the behavior of functions in a more intuitive way. By mastering this skill, you'll gain a deeper appreciation for the power and beauty of mathematics.

So, keep practicing, guys! The more you analyze functions and their graphs, the better you'll become at recognizing patterns and making connections. Before you know it, you'll be a graph-identifying pro!

Remember, the key is to break down the problem into smaller steps, understand the key features of the function, and systematically analyze potential graphs. With a little patience and perseverance, you can conquer any graphing challenge. Now, go forth and graph!

Understanding the Question: Unraveling the Mystery of the Graph

The question at hand, "Which of the following graphs could be the graph of the function f(x) = -0.08x(x^2 - 11x + 18)?" is a classic problem in algebra and precalculus. It tests our ability to connect a function's algebraic representation with its graphical representation. In simpler terms, we need to figure out which picture (graph) matches the equation (function) we're given. This is like matching a face to a name, but in the world of math! The key here is not just blindly guessing, but understanding the properties of functions and how they translate into visual characteristics on a graph. So, let's break down what we need to know to solve this problem.

To successfully identify the correct graph, we need to become detectives of sorts. We'll gather clues from the function's equation and use them to eliminate incorrect graphs until we're left with the right one. This involves understanding several key concepts related to polynomial functions, which include but not limited to: the degree of the polynomial, finding the zeros (or roots) of the function, understanding the end behavior of the graph, and potentially finding some additional points to further narrow down the possibilities. It's like solving a puzzle, where each piece of information helps us fit the bigger picture together. Don't worry if some of these terms sound unfamiliar right now; we'll be explaining them in detail as we go along.

So, guys, let’s keep our focus laser-sharp and prepare to delve into the depths of functions and graphs. This journey might have its twists and turns, but trust me, it's super rewarding once you grasp the fundamentals. We'll approach this step-by-step, making sure each concept is crystal clear before moving on. Think of it as building a solid foundation for your math skills. Are you ready to embark on this adventure? Awesome! Let's roll up our sleeves and get started!

SEO Title: Find the Graph of f(x) = -0.08x(x^2 - 11x + 18) Math Problem Solved

Repair Input Keyword: Which graph represents the function f(x) = -0.08x(x^2 - 11x + 18)?