Identifying The Graph Of F(x) = -0.08x(x² - 11x + 18) A Step-by-Step Guide

Determining the graph of a function can seem like navigating a complex maze, but with a systematic approach, we can unravel the mystery. In this article, we will delve into the intricacies of the function f(x) = -0.08x(x² - 11x + 18) and explore how to identify its corresponding graph. We'll break down the function, analyze its key features, and connect them to the visual representation of the graph. By the end, you'll have a solid understanding of how to approach similar problems and confidently identify the graphs of various functions.

Deconstructing the Function: A Step-by-Step Analysis

To effectively identify the graph of f(x) = -0.08x(x² - 11x + 18), we need to dissect the function and understand its components. This involves factoring the quadratic expression, identifying the roots, analyzing the leading coefficient, and understanding the end behavior. Let's embark on this step-by-step journey:

1. Factoring the Quadratic Expression

The first step in our analysis is to factor the quadratic expression within the function: (x² - 11x + 18). Factoring helps us identify the roots of the function, which are the points where the graph intersects the x-axis. To factor the quadratic, we need to find two numbers that multiply to 18 and add up to -11. These numbers are -2 and -9. Therefore, we can factor the quadratic as follows:

x² - 11x + 18 = (x - 2)(x - 9)

Now, we can rewrite the original function as:

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

This factored form provides valuable insights into the function's behavior.

2. Identifying the Roots

The roots of a function are the values of x for which f(x) = 0. In other words, they are the points where the graph intersects the x-axis. From the factored form of the function, we can easily identify the roots. Setting each factor equal to zero, we get:

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

Therefore, the roots of the function are 0, 2, and 9. These are the x-intercepts of the graph.

3. Analyzing the Leading Coefficient

The leading coefficient is the coefficient of the term with the highest power of x. In this case, after expanding the function, the term with the highest power of x is -0.08x³, so the leading coefficient is -0.08. The leading coefficient plays a crucial role in determining the end behavior of the graph.

A negative leading coefficient indicates that as x approaches positive infinity, f(x) approaches negative infinity, and as x approaches negative infinity, f(x) approaches positive infinity. In simpler terms, the graph will fall to the right and rise to the left.

4. Understanding the End Behavior

End behavior refers to how the graph behaves as x approaches positive and negative infinity. As we discussed in the previous section, the negative leading coefficient dictates the end behavior. For this function, as x approaches positive infinity, the graph will descend towards negative infinity, and as x approaches negative infinity, the graph will ascend towards positive infinity.

5. Determining the Degree of the Polynomial

The degree of the polynomial is the highest power of x in the function. In this case, after expanding the function, the highest power of x is 3, making it a cubic function. Cubic functions have a characteristic S-shape, which we will see in the graph.

By understanding these key features – the roots, the leading coefficient, the end behavior, and the degree of the polynomial – we are well-equipped to identify the graph of the function.

Connecting the Dots: From Function Features to Graph Identification

Now that we have analyzed the function f(x) = -0.08x(x² - 11x + 18) and identified its key features, we can connect these features to the visual representation of the graph. This involves looking for specific characteristics in the graph that correspond to the function's properties.

1. X-Intercepts: The Graph's Anchors

The roots of the function, 0, 2, and 9, are the x-intercepts of the graph. This means the graph will intersect the x-axis at these three points. When examining potential graphs, look for the one that crosses the x-axis at x = 0, x = 2, and x = 9. These points act as anchors for the graph.

2. End Behavior: The Graph's Direction

As we established earlier, the negative leading coefficient dictates the end behavior. The graph should rise to the left (as x approaches negative infinity) and fall to the right (as x approaches positive infinity). This characteristic can help you eliminate graphs with the opposite end behavior.

3. The S-Shape of a Cubic Function

Since the function is a cubic polynomial (degree 3), its graph will have a general S-shape. This shape is characterized by two turning points (local maxima and minima). Look for a graph that exhibits this characteristic curve.

4. Sign Analysis: Mapping the Graph's Trajectory

We can further refine our understanding of the graph by performing a sign analysis. This involves determining the intervals where the function is positive (above the x-axis) and negative (below the x-axis). To do this, we can create a sign chart using the roots as critical points:

This sign chart tells us:

• For x < 0, f(x) is negative (graph is below the x-axis).
• For 0 < x < 2, f(x) is positive (graph is above the x-axis).
• For 2 < x < 9, f(x) is negative (graph is below the x-axis).
• For x > 9, f(x) is positive (graph is above the x-axis).

This information helps us visualize the graph's trajectory and confirm our choice.

5. Identifying Turning Points

Turning points are the points where the graph changes direction (from increasing to decreasing or vice versa). These points correspond to local maxima and minima. While we haven't explicitly calculated the turning points, the S-shape of a cubic function suggests that there will be two turning points. The graph should exhibit these turning points between the roots.

By carefully considering these features – x-intercepts, end behavior, S-shape, sign analysis, and turning points – you can confidently identify the graph of f(x) = -0.08x(x² - 11x + 18) from a set of options.

Example Scenario: Putting the Pieces Together

Let's imagine you are presented with a multiple-choice question asking you to identify the graph of f(x) = -0.08x(x² - 11x + 18) from four different options. Here's how you can apply the knowledge we've gained:

1. Eliminate graphs that don't have x-intercepts at 0, 2, and 9. This immediately narrows down the possibilities.
2. Eliminate graphs that don't exhibit the correct end behavior. The graph should rise to the left and fall to the right. This further reduces the options.
3. Look for the graph with the characteristic S-shape of a cubic function. This helps you identify the general form of the graph.
4. Use the sign analysis to confirm the graph's trajectory. The graph should be below the x-axis for x < 0, above the x-axis for 0 < x < 2, below the x-axis for 2 < x < 9, and above the x-axis for x > 9.
5. Consider the presence of turning points. The graph should have two turning points, reflecting the cubic nature of the function.

By systematically applying these steps, you can confidently select the correct graph.

Conclusion: Mastering the Art of Graph Identification

Identifying the graph of a function is a fundamental skill in mathematics. By understanding the relationship between a function's algebraic properties and its visual representation, we can confidently navigate the world of graphs. In this article, we have meticulously analyzed the function f(x) = -0.08x(x² - 11x + 18), breaking it down into its key components and connecting those components to the characteristics of its graph.

We explored factoring, roots, leading coefficients, end behavior, and the degree of the polynomial. We then used this information to identify x-intercepts, predict the graph's direction, recognize the S-shape of a cubic function, and perform a sign analysis. By mastering these techniques, you can confidently approach similar problems and unlock the secrets hidden within the graphs of functions.

Remember, practice is key. The more you analyze functions and their graphs, the more intuitive this process will become. So, embrace the challenge, explore the world of functions, and let the graphs reveal their stories to you.

Additional Tips for Graph Identification

To further enhance your graph identification skills, consider these additional tips:

1. Use graphing calculators or software: These tools can help you visualize the graph of a function and verify your analysis.
2. Practice with a variety of functions: The more functions you analyze, the better you'll become at recognizing patterns and connecting algebraic properties to graphical features.
3. Pay attention to scale: The scale of the axes can sometimes be misleading. Be sure to carefully consider the scale when interpreting a graph.
4. Look for symmetries: Some functions exhibit symmetry (e.g., even functions are symmetric about the y-axis, and odd functions are symmetric about the origin). Recognizing symmetry can help you identify the graph.
5. Consider transformations: Understanding how transformations (e.g., shifts, stretches, reflections) affect the graph of a function can be a powerful tool.

By incorporating these tips into your practice, you'll be well on your way to becoming a graph identification expert.