Graphing and Analyzing y2x3

Graphing a function such as y2x3 involves a series of steps that help us visualize the relationship between x and y. This guide will walk you through the process, including finding key points, understanding the function's behavior, and drawing the graph.

Understanding the Function

The given function is y2x3. This is a cubic function, which means it is a polynomial of degree 3. Unlike linear functions (where y mx b) or quadratic functions (where y ax2 bx c), cubic functions can have more complex shapes, including turning points and inflection points.

Key Points and Basic Analysis

To graph the function, we can start by determining where the curve intersects the x-axis and y-axis. These points are often referred to as the x-intercept and y-intercept, respectively.

Y-Intercept

First, we determine the y-intercept by setting x0. Plugging x0 into the equation:

y 2(0)3 y 0

Therefore, the curve intersects the y-axis at the point (0,0).

X-Intercept

To find the x-intercept, set y0 and solve for x:

0 2x3

Dividing both sides by 2:

0 x3

Solving for x:

x 0

Thus, the curve intersects the x-axis at the point (0,0).

Plotting Additional Points

While the intercepts give us a starting point, we will need more points to draw a smooth curve. Let's calculate the value of y for a couple of additional x-values.

When x 1:

y 2(1)3 y 2

When x -1:

y 2(-1)3 y -2

These points are (1, 2) and (-1, -2).

Graphing the Function

Now, plot the points (0,0), (1, 2), and (-1, -2) on a coordinate plane. Since y2x3 is a cubic function, the curve will start from the bottom left (negative x and y), pass through the origin, and end in the top right (positive x and y). The curve will be smooth and continuous, with no breaks or sharp corners.

Function Analysis

A cubic function like y2x3 has an inflection point, which is a point where the curve changes its concavity. For y2x3, the inflection point occurs at x 0. This is where the curve changes from concave down to concave up.

The function is defined for all real numbers (x ∈ R). To understand the range of y values, we note that as x increases or decreases without bound, y also increases or decreases without bound, respectively. This means the function covers all real numbers (y ∈ R).

Conclusion

In conclusion, the graph of y2x3 is a smooth, cubic curve that passes through the origin. It is symmetrical about the origin, and its range is all real numbers. While the function is not one-to-one (since it has multiple y values for some x values), its graph provides a clear visual representation of the relationship between x and y.

For a more complete understanding, one can consider the mirror image of a portion of the graph for x 0 and x 0 about the x-axis, as described previously, to reinforce the symmetry.