What does the graph of the polynomial function f(x) = 4x^7 – 40x^6 + 100x^5 look like?

by

The polynomial function f(x) = 4x7 – 40x6 + 100x5 is characterized by its degree, which is the highest power of x present in the expression. In this case, the degree is 7, indicating that the graph will have a series of turning points and will extend to infinity in both positive and negative directions.

To better understand the graph, let’s analyze the key features:

  • Leading Coefficient: The leading coefficient is 4 (from the term 4x7), which is positive. This means that as x approaches positive infinity, f(x) will also approach positive infinity, and as x approaches negative infinity, f(x) will approach negative infinity.
  • Zeros of the Function: To find the x-intercepts (where the graph crosses the x-axis), we can set f(x) to zero. Factoring out the common term can help simplify the problem:
    • f(x) = x5(4x2 - 40x + 100)
  • This indicates that there are at least 5 x-intercepts at x = 0. To find additional roots, we can analyze the quadratic 4x2 – 40x + 100.
  • Using the quadratic formula, we can determine the additional roots:
    • x = [40 ± √(402 - 4(4)(100) ] / (2*4)
  • Solving this will provide the additional x-intercepts beyond x = 0.

End Behavior: The graph behaves as follows:

  • As x → ∞, f(x) → ∞
  • As x → -∞, f(x) → -∞

Turning Points: Since the function degree is 7, it can have up to 6 turning points.

Overall, the graph of f(x) = 4x7 – 40x6 + 100x5 will display a complex pattern typical of higher-degree polynomial functions, with multiple intercepts and areas of increase and decrease, providing a rich visual representation of polynomial behavior.

Leave a Comment