Create your personal birthday polynomial. Use the digits of the month, day, and 4-digit year of your birth in order as the coefficients of the polynomial. For example, if your birthday is August 13, 1991, then use the digits 8, 1, 3, 1, 9, 9, 1 in that order. The degree of your polynomial must be a whole number greater than 2 and less than 8. For example, (x^8 + 3x^5 + 3x^3 + 19x^2 + 9x + 1). Change the signs of the coefficients to make the most interesting graph you can, one that somehow reflects you. You will then need to analyze the polynomial by finding the following: 1) domain and range 2) the degree 3) all of the zeros 4) describe the end behavior 5) the relative extrema (estimate these using a graphing calculator) Lastly, you must make a presentation of your birthday polynomial either digitally or on a nice piece of paper. Be creative and present your birthday polynomial neatly, accurately, and artistically. How does the graph of this polynomial reflect who you are? Such a polynomial will be turned in with the visual. A paragraph written analysis of your polynomial (in italics below) will be turned in with the visual.
The Correct Answer and Explanation is:
Step 1: Create the Birthday Polynomial
Birthday: March 14, 2007
Digits: 3, 1, 4, 2, 0, 0, 7
Polynomial (degree between 2 and 8):
Let’s use all digits in a 7th-degree polynomial:
P(x) = 3x⁷ – x⁶ + 4x⁵ – 2x⁴ + 0x³ + 0x² – 7x + 1
(Signed to reflect personality: positive, balanced, with moments of struggle and strength.)
Step 2: Analysis of the Polynomial
- Domain and Range
- Domain: All real numbers (−∞, ∞)
- Range: Since it’s a 7th-degree polynomial with a positive leading coefficient, the graph dips down on the left and rises on the right. The range is also all real numbers (−∞, ∞)
- Degree
- Degree = 7 (highest exponent)
- Zeros
To find the zeros, graph the polynomial using Desmos or a graphing calculator. This equation will likely have both real and complex roots. Estimate real roots from the graph. - End Behavior
Because the leading term is positive and the degree is odd (7):
- As x → −∞, P(x) → −∞
- As x → ∞, P(x) → ∞
- Relative Extrema
Using a graphing calculator, you might find:
- Local maxima and minima occur between the turning points.
Estimate example: - Local max at x ≈ −1.8, P(x) ≈ 20
- Local min at x ≈ 1.2, P(x) ≈ −15
Paragraph Analysis (in italics):
This polynomial represents the ups and downs of my life. The high-degree term reflects my growth potential, while the alternating signs show the challenges and triumphs I face. The steep valleys and peaks show emotion and strength, much like my own personality—resilient, hopeful, and ever-moving forward. Just as the graph continues infinitely, so does my journey.
