The function f(x) = 3x4 + 1 represents a polynomial curve where the highest degree term is x raised to the power of 4. Here are some key characteristics and steps to understand its graph:
1. Basic Formula
The polynomial function consists of a leading coefficient of 3 and a constant term of 1. This means that the graph will be shaped largely by the x4 term, as it will dominate the behavior of the function for large values of x (both positively and negatively).
2. End Behavior
Because the degree of the polynomial is even (4), the ends of the graph will behave the same way in both directions:
- As x approaches positive infinity (x → +∞), f(x) will also approach positive infinity (f(x) → +∞).
- As x approaches negative infinity (x → -∞), f(x) will approach positive infinity as well (f(x) → +∞).
3. Vertex and Minimum Point
The function has no real roots; instead, it will have a minimum point due to the positive leading coefficient. The minimum value occurs when x = 0:
- f(0) = 3(0)4 + 1 = 1
This means the point (0, 1) is the lowest point on the graph.
4. Axis of Symmetry
The function is symmetric about the y-axis because it is an even function. For every point (x, y) on the graph, there is a corresponding point (-x, y).
5. Visual Representation
To graph this function:
- Start with the vertex at (0, 1).
- As you move away from the vertex (in both directions), the values of f(x) will increase rapidly due to the x4 term.
- Thus, the graph will rise steeply in both directions from (0, 1).
6. Summary
In conclusion, the graph of f(x) = 3x4 + 1 will show a U-shaped curve that has:
- A minimum point at (0, 1)
- Symmetry about the y-axis
- Ends that rise towards infinity as you move away from the minimum.
You can visualize this function using graphing software or tools to see these characteristics illustrated clearly.