Understanding the Graph of g(x) = 0.5x³ + 4
The function g(x) = 0.5x³ + 4 is a cubic function characterized by its unique S-shaped curve, known for its turning points and end behavior. Let’s break down its components and how it influences the graph.
Key Features of the Function
- Leading Coefficient: The leading coefficient is 0.5, which means the graph will rise to the right and fall to the left. This is typical for cubic functions where the leading coefficient is positive.
- y-Intercept: The y-intercept occurs when x = 0. Plugging this value into the function gives g(0) = 0.5(0)³ + 4 = 4. Therefore, the graph crosses the y-axis at (0, 4).
- End Behavior: As x approaches positive infinity, g(x) tallies to positive infinity as well, and conversely, as x approaches negative infinity, g(x) will also trend down towards negative infinity.
Graphing g(x) = 0.5x³ + 4
To sketch the graph:
- Start with the y-intercept at (0, 4).
- Determine additional points by choosing values for x. For instance:
- g(1) = 0.5(1)³ + 4 = 4.5
- g(-1) = 0.5(-1)³ + 4 = 3.5
- g(2) = 0.5(2)³ + 4 = 8
- g(-2) = 0.5(-2)³ + 4 = 0
- Plot these points on the graph.
- Notice the curvature between these points, reflecting the S-like shape of a cubic function.
Conclusion
In summary, the graph of g(x) = 0.5x³ + 4 is a smooth curve that demonstrates the typical behavior of a cubic function. With its distinct turning points and the shift upwards by 4 units due to the constant term, it presents a visually appealing depiction of polynomial growth.