Graphing The Function Y = -5/4x³ A Step-by-Step Guide

In the realm of mathematics, understanding the behavior of functions is crucial. Graphing functions allows us to visualize their properties and relationships, providing valuable insights into their nature. In this article, we will delve into the process of graphing the cubic function y = -5/4x³. This function, a member of the polynomial family, exhibits unique characteristics that we will explore through plotting points and analyzing its graph.

Before we embark on graphing the specific function, it is essential to grasp the fundamental nature of cubic functions. Cubic functions are polynomial functions of degree three, meaning the highest power of the variable 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 is not equal to zero. The leading coefficient, a, plays a significant role in determining the function's end behavior. If a is positive, the graph rises to the right and falls to the left. Conversely, if a is negative, the graph falls to the right and rises to the left. Cubic functions can have up to three real roots (x-intercepts) and can have at most two turning points (local maxima or minima). The symmetry of a cubic function is also noteworthy. They exhibit rotational symmetry about a point of inflection, which is the point where the concavity changes.

The function we are exploring, y = -5/4x³, is a special case of a cubic function where b, c, and d are zero. This simplified form allows us to focus on the impact of the leading coefficient, -5/4, on the graph's shape and orientation. The negative leading coefficient indicates that the graph will fall to the right and rise to the left. Additionally, the absence of other terms implies that the graph will pass through the origin (0, 0) and exhibit symmetry about the origin.

To graph the function y = -5/4x³, we will employ the method of plotting points. This involves selecting various x-values, substituting them into the function, and calculating the corresponding y-values. These pairs of (x, y) values represent points on the graph. By plotting a sufficient number of points, we can discern the shape and trend of the function. For this exercise, we will plot five points, including one point with x = 0, two points with negative x-values, and two points with positive x-values. This selection will provide a balanced representation of the function's behavior across the coordinate plane.

Let's start with x = 0. Substituting this value into the function, we get:

y = -5/4(0)³ = 0

Thus, the first point is (0, 0), which is the origin. Next, let's choose two negative x-values, such as x = -2 and x = -1. For x = -2:

y = -5/4(-2)³ = -5/4(-8) = 10

This gives us the point (-2, 10). For x = -1:

y = -5/4(-1)³ = -5/4(-1) = 5/4 = 1.25

So, the second point is (-1, 1.25). Now, let's select two positive x-values, such as x = 1 and x = 2. For x = 1:

y = -5/4(1)³ = -5/4(1) = -5/4 = -1.25

This yields the point (1, -1.25). For x = 2:

y = -5/4(2)³ = -5/4(8) = -10

This gives us the point (2, -10). We now have five points: (0, 0), (-2, 10), (-1, 1.25), (1, -1.25), and (2, -10). These points will serve as the foundation for graphing the function.

With the five points calculated, we can now plot them on a coordinate plane. The x-axis represents the input values, and the y-axis represents the output values. Plot each point by locating its x-coordinate on the x-axis and its y-coordinate on the y-axis, then marking the intersection. Once all points are plotted, we can connect them with a smooth curve to create the graph of the function. The smooth curve should pass through all the plotted points, reflecting the function's behavior between these points. As we connect the points, we should keep in mind the general shape of a cubic function with a negative leading coefficient, which falls to the right and rises to the left. The curve will pass through the origin (0, 0) and exhibit symmetry about the origin. The graph will extend infinitely in both directions, illustrating the function's behavior for all real numbers.

After plotting the points (0, 0), (-2, 10), (-1, 1.25), (1, -1.25), and (2, -10) and connecting them with a smooth curve, we will observe the characteristic S-shape of a cubic function. The graph will rise steeply to the left of the origin and fall steeply to the right of the origin, reflecting the negative leading coefficient of -5/4. The symmetry about the origin will be evident, as the graph on one side of the origin is a mirror image of the graph on the other side. The graph will demonstrate that as x becomes increasingly negative, y becomes increasingly positive, and as x becomes increasingly positive, y becomes increasingly negative. This visual representation of the function allows us to understand its behavior and properties more intuitively.

Once the graph is plotted, we can analyze its key features. The intercepts, where the graph crosses the x-axis and y-axis, provide valuable information about the function's roots and behavior near zero. The turning points, which are the local maxima and minima, indicate where the function changes direction. The end behavior, which describes the function's behavior as x approaches positive and negative infinity, reveals the function's long-term trend. In the case of y = -5/4x³, the graph has only one intercept, which is at the origin (0, 0). This means that the function has only one real root, which is x = 0. The graph does not have any turning points, indicating that the function is always either increasing or decreasing. As we discussed earlier, the end behavior is such that the graph rises to the left and falls to the right, consistent with the negative leading coefficient.

The symmetry of the graph is another important aspect to consider. As mentioned earlier, the graph of y = -5/4x³ exhibits rotational symmetry about the origin. This means that if we rotate the graph 180 degrees about the origin, it will coincide with itself. This symmetry is a characteristic feature of odd functions, which are functions that satisfy the condition f(-x) = -f(x). The function y = -5/4x³ is an odd function, as can be verified by substituting -x into the function: y = -5/4(-x)³ = -5/4(-x³) = 5/4x³ = -(-5/4x³). This confirms that the function is odd and its graph is symmetric about the origin.

Graphing the function y = -5/4x³ provides a visual representation of its behavior and properties. By plotting points and connecting them with a smooth curve, we can observe the characteristic S-shape of a cubic function with a negative leading coefficient. The graph rises to the left and falls to the right, passes through the origin, and exhibits symmetry about the origin. Analyzing the graph allows us to identify key features such as intercepts, turning points, and end behavior. The function has one intercept at the origin, no turning points, and its end behavior is such that it rises to the left and falls to the right. The graph's symmetry about the origin is a consequence of the function being an odd function. This exercise demonstrates the power of graphing functions as a tool for understanding their mathematical properties and relationships.

Graphing functions is a fundamental skill in mathematics, and the process of plotting points and analyzing the resulting graph provides valuable insights into the function's behavior. The cubic function y = -5/4x³ serves as an excellent example to illustrate this process. By understanding the concepts and techniques discussed in this article, you can confidently graph and analyze a wide range of functions, enhancing your mathematical understanding and problem-solving abilities.