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Exploring the Vertex and Graph of the Function f(x) x^8 - 3

May 09, 2025Technology1292
Exploring the Vertex and Graph of the Function f(x) x8 - 3 The functi

Exploring the Vertex and Graph of the Function f(x) x8 - 3

The function f(x) x^8 - 3 is a polynomial function that opens upwards, and its graphical representation can be quite intriguing. By analyzing its vertex and plotting its graph, we can gain a deeper understanding of its properties and behavior.

Understanding the Vertex of f(x) x8 - 3

The vertex of the function f(x) x^8 - 3 is found at (0, -3). This point is significant because it represents the minimum value of the function, given that the term x^8 always results in a non-negative value for any real number x.

Key Observations

When x 0, the function simplifies to:

f(0) 0^8 - 3 -3

Thus, the vertex lies at the point (0, -3). This is the point where the function f(x) x^8 - 3 reaches its lowest value, illustrating that it opens upwards, corresponding to the shape of a regular trapezoid.

The Graph of f(x) x8 - 3

The graph of the function y f(x) x^8 - 3 can be visualized by plotting various points. From the table given:

x y 0 -3 -1 4 -2 3 0 -3 -5 0 -11 0

Plotting these points, we can observe that the graph consists of two straight lines:

Line I: y x^8 - 3

Line II: y -x^8 - 3

Line I: This line appears for x -8 or x 8. However, for x ≠ -8 or x ≠ 8, the line is defined as y x^8 - 3. Line II: This line appears for x ≠ -8 or x ≠ 8. For x -8 or x 8, the line is defined as y -x^8 - 3.

Intersection Points and Continuity

The two lines intersect each other at the point where x -8 or x 8. At these points, the function simplifies as follows:

f(-8) (-8)^8 - 3 16777216 - 3

f(8) 8^8 - 3 16777216 - 3

Both of these points result in the same value, showcasing the symmetry of the function.

From the table, we can see that the function values at these points are:

f(-8) 0 - 3 -3

f(8) 0 - 3 -3

Therefore, the lines intersect at the point (-8, -3) and (8, -3), which is the vertex of the function, as mentioned earlier.

Conclusion

The function f(x) x^8 - 3 has a vertex at (0, -3). Its graph consists of two straight lines that intersect at the vertex and are defined by different equations based on the value of x. This function is an excellent example of how polynomial functions can be analyzed and visualized for deeper understanding.