The Birthday Polynomial Project Due date: Wednesday 12/09/20 at the beginning of class (10 points deduction for each day late). The project is worth one grade in Algebra 2. Create a Birthday Polynomial. Use the digits of the month, day, and four (4) digit year of your birth in order as the coefficients of the polynomial: (For example: If your birthday is February 5th, 2019, then use the digits 252019 in that order). The degree of your polynomial must be a whole number greater than 2 and less than 6. (Example: y = 2x^4 + 8x^3 + 20x + 19). Use only one variable throughout the polynomial (1 letter). Change the signs of the coefficients to make the most interesting graph you can; one that in some way reflects you. Analyze the Polynomial: 1. Find the y-intercept (x-intercept). 2. Classify the degree, leading term, and leading coefficient of the polynomial. 3. Classify the type of polynomial: monomial, binomial, etc. (If other, state how many terms there are). Identify relative minimums, relative maximums, and end behavior. 4. Graph the polynomial on coordinate grid paper (Desmos Graphing calculator). Find five ordered pairs (x, y) that satisfy your polynomial (use an input-output table).
The Correct Answer and Explanation is:
Sample Birthday: March 14, 2005
Digits: 3 (March), 14 (day), 2005 (year) → Combine as: 3142005
Let’s pick a degree of 5 (between 3 and 5 as required).
Step 1: Create the polynomial
Use digits 3, 1, 4, 2, 0, 0, 5 as coefficients. Let’s arrange them to create an interesting polynomial, possibly changing signs.
Let:
y = -3x⁵ + x⁴ – 4x³ + 2x² + 0x – 5
(We can ignore the 0x since it contributes nothing.)
Step 2: Analyze the Polynomial
- Y-intercept (when x = 0):
y = -3(0)⁵ + (0)⁴ – 4(0)³ + 2(0)² – 5 = -5
So, the y-intercept is (0, -5) - Degree: 5
Leading term: -3x⁵
Leading coefficient: -3 - Type of polynomial:
It has 5 terms: a polynomial with 5 terms (not mono-, bi-, or trinomial) - Relative minimums and maximums:
Use Desmos or a graphing calculator to find these turning points. - End behavior:
Since the degree is odd and the leading coefficient is negative:- As x → ∞, y → -∞
- As x → -∞, y → ∞
Step 3: Find five ordered pairs (input-output table)
| x | y |
|---|---|
| -2 | -3(-32) + 16 + 32 + 8 – 5 = 87 |
| -1 | 3 – 1 + 4 + 2 – 5 = 3 |
| 0 | -5 |
| 1 | -3 + 1 – 4 + 2 – 5 = -9 |
| 2 | -96 + 16 – 32 + 8 – 5 = -109 |
Explanation
The Birthday Polynomial Project allows students to apply personal information in a mathematical context by building a polynomial using their birthdate. This project uses the digits from a fictional birthday—March 14, 2005—to form the polynomial: y = -3x⁵ + x⁴ – 4x³ + 2x² – 5. This polynomial has a degree of 5, which meets the requirement of being greater than 2 and less than 6. By changing the signs of the coefficients, we create a graph that has varying peaks and valleys, making it more expressive and complex.
Analyzing the polynomial helps understand its behavior. The y-intercept is found by substituting 0 for x, resulting in a value of -5. This tells us the graph crosses the y-axis at the point (0, -5). The degree is 5, so the leading term is -3x⁵ and the leading coefficient is -3. Since it has five terms, it is simply called a polynomial with five terms, not a monomial, binomial, or trinomial.
The end behavior is determined by the degree and the sign of the leading coefficient. Because the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right. That is, as x becomes very large, y decreases, and as x becomes very negative, y increases.
To deepen the understanding, an input-output table was created to find ordered pairs that satisfy the equation. Plotting these points using a graphing tool like Desmos gives a visual representation of how the polynomial behaves. This project connects abstract algebraic concepts to real-world personal data and offers a hands-on approach to learning about polynomials.
