Graphing Y = -3/2 X³ A Step-by-Step Guide

In the realm of mathematics, understanding how to graph functions is a fundamental skill. Graphing allows us to visualize the behavior of a function, identify key features, and make predictions about its output for different inputs. In this comprehensive guide, we will delve into the process of graphing the function y = -3/2 x³. This particular function is a cubic function, characterized by its distinctive S-shaped curve. We will explore the properties of cubic functions, discuss the step-by-step method for graphing y = -3/2 x³, and highlight the key features of the graph. By the end of this guide, you will have a solid understanding of how to graph cubic functions and interpret their graphical representation.

Before we dive into graphing y = -3/2 x³, let's first establish a firm understanding of cubic functions in general. A cubic function is a polynomial function of degree three, meaning the highest power of the variable (in this case, x) is three. The general form of a cubic function is:

f(x) = ax³ + bx² + cx + d

where a, b, c, and d are constants, and a ≠ 0. The coefficient a plays a crucial role in determining the overall shape and direction of the cubic function's graph.

Key Characteristics of Cubic Functions:

1. Domain and Range: Cubic functions have a domain and range of all real numbers, meaning they can accept any real number as input and produce any real number as output.
2. End Behavior: The end behavior of a cubic function describes what happens to the function's graph as x approaches positive or negative infinity. For cubic functions with a positive leading coefficient (a > 0), the graph rises to the right (as x approaches positive infinity) and falls to the left (as x approaches negative infinity). Conversely, for cubic functions with a negative leading coefficient (a < 0), the graph falls to the right and rises to the left. In our case, y = -3/2 x³ has a negative leading coefficient (-3/2), so it will fall to the right and rise to the left.
3. Symmetry: Cubic functions can exhibit either odd symmetry (symmetry about the origin) or no symmetry. A cubic function has odd symmetry if f(-x) = -f(x) for all x. The function y = -3/2 x³ exhibits odd symmetry.
4. Inflection Point: A cubic function has exactly one inflection point, which is the point where the concavity of the graph changes. The concavity of a graph refers to whether it is curving upwards (concave up) or downwards (concave down). For y = -3/2 x³, the inflection point is at the origin (0, 0).
5. Roots (Zeros): A cubic function can have up to three real roots, which are the values of x where the function intersects the x-axis (i.e., where y = 0). The function y = -3/2 x³ has one real root at x = 0.

Understanding these characteristics of cubic functions is essential for effectively graphing them. Now, let's move on to the specific function y = -3/2 x³ and explore how to graph it step by step.

To graph the function y = -3/2 x³, we will follow a systematic approach that involves creating a table of values, plotting the points, and connecting them to form the graph. Here's the step-by-step method:

Step 1: Create a Table of Values

Start by selecting a range of x-values that will give you a good representation of the function's behavior. It's helpful to include both positive and negative values, as well as zero. For y = -3/2 x³, we can choose the following x-values:

Calculate the corresponding y-values for each x-value using the function y = -3/2 x³. For example, when x = -2, we have:

y = -3/2 * (-2)³ = -3/2 * (-8) = 12

Repeat this calculation for all the chosen x-values and fill in the table.

Step 2: Plot the Points

Now, use the x and y values from the table to plot the points on a coordinate plane. Each pair of (x, y) values represents a point on the graph. For example, the first point from our table is (-2, 12), so we would plot a point at the location where x = -2 and y = 12.

Plot all the points from your table onto the coordinate plane. Make sure to use a consistent scale for both the x-axis and the y-axis.

Step 3: Connect the Points

Once you have plotted all the points, connect them with a smooth curve. Remember that cubic functions have a characteristic S-shape, so your curve should reflect this. Since the leading coefficient of y = -3/2 x³ is negative, the graph will rise to the left and fall to the right. Also, keep in mind that the function has odd symmetry, meaning it is symmetric about the origin.

As you connect the points, pay attention to the inflection point, which is at (0, 0) for this function. The concavity of the graph will change at this point. To the left of the inflection point, the graph is concave down, and to the right of the inflection point, the graph is concave up.

By following these three steps, you can accurately graph the function y = -3/2 x³. The resulting graph will be a smooth, S-shaped curve that passes through the plotted points and exhibits the characteristics of a cubic function with a negative leading coefficient.

Now that we have graphed the function y = -3/2 x³, let's take a closer look at its key features. Identifying these features can provide valuable insights into the function's behavior and properties.

1. Shape: The graph of y = -3/2 x³ is a cubic curve with an S-shape. This shape is characteristic of cubic functions, which have a point of inflection where the concavity changes.
2. End Behavior: As we discussed earlier, the end behavior of a cubic function is determined by the sign of its leading coefficient. Since the leading coefficient of y = -3/2 x³ is negative (-3/2), the graph rises to the left (as x approaches negative infinity) and falls to the right (as x approaches positive infinity). This is evident in the graph, which extends upwards on the left side and downwards on the right side.
3. Symmetry: The graph of y = -3/2 x³ exhibits odd symmetry, also known as symmetry about the origin. This means that if you rotate the graph 180 degrees about the origin, it will look the same. Mathematically, this is expressed as f(-x) = -f(x). For example, if you plug in x = 2, you get y = -12, and if you plug in x = -2, you get y = 12. The points (2, -12) and (-2, 12) are symmetric with respect to the origin.
4. Inflection Point: The inflection point of the graph is the point where the concavity changes. For y = -3/2 x³, the inflection point is at the origin (0, 0). To the left of the origin, the graph is concave down, meaning it curves downwards. To the right of the origin, the graph is concave up, meaning it curves upwards. The inflection point marks the transition between these two concavities.
5. Roots (Zeros): The roots or zeros of a function are the values of x where the graph intersects the x-axis (i.e., where y = 0). For y = -3/2 x³, there is only one real root, which is at x = 0. This means that the graph touches the x-axis only at the origin.

Understanding these key features allows you to quickly analyze and interpret the graph of y = -3/2 x³. By recognizing the shape, end behavior, symmetry, inflection point, and roots, you can gain a deeper understanding of the function's behavior.

Now that we have explored the graph of the basic cubic function y = -3/2 x³, let's briefly discuss how transformations can affect the graph of a cubic function. Transformations are operations that shift, stretch, compress, or reflect a graph, and they can provide valuable insights into the relationship between different functions.

The general form of a transformed cubic function is:

y = a(x - h)³ + k

where a, h, and k are constants that control the transformations.

1. Vertical Stretch/Compression and Reflection: The constant a controls the vertical stretch or compression and reflection of the graph. If |a| > 1, the graph is vertically stretched, making it steeper. If 0 < |a| < 1, the graph is vertically compressed, making it flatter. If a < 0, the graph is reflected across the x-axis. In the case of y = -3/2 x³, the value a = -3/2 indicates a vertical stretch (since |-3/2| > 1) and a reflection across the x-axis (since a is negative).
2. Horizontal Shift: The constant h controls the horizontal shift of the graph. If h > 0, the graph is shifted to the right by h units. If h < 0, the graph is shifted to the left by |h| units. For example, the graph of y = (x - 2)³ would be the same as the graph of y = x³ shifted 2 units to the right.
3. Vertical Shift: The constant k controls the vertical shift of the graph. If k > 0, the graph is shifted upwards by k units. If k < 0, the graph is shifted downwards by |k| units. For example, the graph of y = x³ + 3 would be the same as the graph of y = x³ shifted 3 units upwards.

By understanding these transformations, you can quickly sketch the graph of a transformed cubic function by starting with the basic cubic function and applying the appropriate shifts, stretches, compressions, and reflections.

In this comprehensive guide, we have explored the process of graphing the function y = -3/2 x³. We began by understanding the general characteristics of cubic functions, including their domain, range, end behavior, symmetry, inflection point, and roots. We then followed a step-by-step method to graph y = -3/2 x³, which involved creating a table of values, plotting the points, and connecting them to form a smooth curve. We also identified the key features of the graph, such as its shape, end behavior, symmetry, inflection point, and roots. Finally, we discussed how transformations can affect the graph of a cubic function.

By mastering the techniques and concepts presented in this guide, you will be well-equipped to graph cubic functions and interpret their graphical representations. Graphing functions is a fundamental skill in mathematics, and it provides a powerful tool for visualizing and understanding the behavior of mathematical relationships. With practice and a solid understanding of the principles involved, you can confidently graph a wide range of functions and gain valuable insights into their properties.

This knowledge extends beyond the classroom, as the ability to interpret graphical data is crucial in various fields, including science, engineering, economics, and data analysis. Continue to explore different types of functions and practice graphing them to further enhance your mathematical skills and analytical abilities.