Polynomial Functions Unit Assignment

Question 1: Analysis of Polynomial Functions

Function Analysis Chart

End Behaviours

• For polynomial functions, the end behaviours depend on the leading term:

  • Even degree leading term (e.g., (x^4), (x^6)): Both ends rise or both ends fall.

  • Odd degree leading term (e.g., (x^7)): One end rises, and the other end falls.

Maximum and Minimum Number of x-Intercepts

• Maximum x-intercepts: Equal to the degree of the polynomial.

• Minimum x-intercepts: Can be 0.

Maximum and Minimum Number of Turns

• Maximum turns: Equal to degree - 1.

• Minimum turns: Can be 0 (at maximum or minimum points).

Restrictions on Domain and Range

• Domain of all polynomial functions: ( (-\infty, \infty) ).

• Range of polynomial functions depends on the leading coefficient and the value of k

Question 2: Finding Polynomial Functions from Graphs

Instructions

• Examine the provided graphs for the polynomial functions. Identify key points and characteristics to formulate equations. Show all work.

Question 3: Modelling Polynomial Data

Instructions

• Based on the provided table of values.

• Identify trends and create polynomial equations that model the data.

• Determine the symmetry of given polynomial functions:

  • Even Symmetry:( f(-x) = f(x) )

  • Odd Symmetry: ( f(-x) = -f(x) )

  • Neither: If neither condition is met.

Functions for Symmetry Analysis

• a) ( f(x) = -x^3 + 8x + 12 )

• b) ( f(x) = 3x^4 - 7x^8 )

• c) ( f(x) = \frac{1}{2}x^5 + \frac{1}{3}x^3 )

• d) ( f(x) = 4x^3 + x^2 )

Factorization

• Factor each polynomial function completely and show work.

Question 4: Real-World Modelling with Polynomials

Ice Thickness Model

• Given function: ( T(d) = -0.1d^3 + 1.2d^2 - 4.4d + 14.8 )

  • T: Thickness in cm

  • d: Days after December 31st

Profit Model

• Given function: ( P(t) = -4t^3 + 10t^2 + 8t - 6 )

  • P: Profit in thousands of dollars

  • t: Time in years

Questions to Answer

• When will maximum profit of $18,000 occur?

Question 5: Graphing and Analysis

Instructions

a) Graph the function using technology, ensuring axes are correctly set.
b) State the domain and range for the situation.

Warmest Day Estimation

• Determine during which day the warmest temperature occurred. Justify the answer using the graph.

Average Rate of Change

• Calculate average rate of change on a short interval near the chosen point.

Instantaneous Rate of Change

• Determine instantaneous rate of change at the chosen point.

Comparison of Rates

• Discuss whether rates from a) and b) are the same or different and provide reasoning.

Submission Information

• Complete all questions and upload work to the specified dropbox.

As a text-based AI, I cannot generate image files directly; however, I can provide the critical data points and a visual description to help you sketch or plot the graph for Question 5. Assuming the function for the temperature/thickness situation is $T(d) = -0.1d^3 + 1.2d^2 - 4.4d + 14.8$, here is the graph analysis:

Graph Analysis for $T(d) = -0.1d^{3} + 1.2d^{2} - 4.4d + 14.8$

1. Key Points for Plotting:

   • $y$-intercept: At $d = 0$, $T(0) = 14.8$. Point: $(0, 14.8)$.

   • Local Minimum: Occurs at approximately $d \approx 2.85$. Point: $(2.85, 9.53)$.

   • Local Maximum (Warmest Day Potential): Occurs at approximately $d \approx 5.15$. Point: $(5.15, 10.27)$.

2. Shape and End Behaviour:

   • Since the function is a cubic with a negative leading coefficient ($-0.1d^3$):

     • As $d \to \infty$, $T(d) \to -\infty$.

     • As $d \to -\infty$, $T(d) \to \infty$.

   • Within the relevant domain (typically positive $d$), the curve starts high, dips to a low point around day 3, rises to a peak around day 5, and then falls again.

3. Visual Representation (ASCII Sketch):

   T (Value)
   ^
   |  * (0, 14.8)
   |   \
   |    \
   |     \             * (5.15, 10.27) <--- Warmest Day peak
   |      \           / \
   |       *---------*   \
   |    (2.85, 9.53)      \
   +---------------------------> d (Days)

To see an accurate digital rendering, you can enter the equation $y = -0.1x^3 + 1.2x^2 - 4.4x + 14.8$ into a graphing calculator like Desmos.