PrecalculusResearch writing prompt

This is a mathematical research paper. You will respond to the prompts given below in paragraph form using complete, organized sentences, in 3-5 pages. You may use outside sources, but they must be cited correctly with a works cited page. Your paper must be typed with Times New Roman, 12-point font, double-spaced. Any plagiarism will result in a zero.

First rough draft is due Thursday, April 24. Final copy is due Wednesday, April 30.First rough draft must have an introduction and two additional paragraphs in answer to the seed question and at least two other questions/topics. It can be hand-written.

Choose one of the following prompts:

InterceptsSeed question:

What are the x- and y-intercepts of the equation y=x2−5 x+6?Questions to build on:

What are the x- and y-intercepts of the equation y=x2+9? What is the maximum number of x-intercepts that a linear equation can

have? Quadratic equation? Cubic? Quartic? Nth-degree polynomial equation? Trigonometric?

What is the minimum number of x-intercepts an equation can have? How many y-intercepts can different equations have?

Extension questions: What are the x-, y-, and z-intercepts of the equation z= y2+4 x+6 y−3 xy+8? What are the x- and y-intercepts of a periodic equation like y=sin x or

y=cos x? Derive a formula for the x- and y-intercepts of all linear equations. Derive a formula for the x- and y-intercepts of all quadratic equations. Are

there some equations without any x-intercepts? Y-intercepts?

Ancient number systemsSeed question:

What are some of the first number systems, and how do they differ from our current decimal, place-value system?

Questions to build on: What are the symbols used by the different ancient number systems? Rewrite and compute the equations using Egyptian, Babylonian, and Roman

numerals:o 3+15=¿o 100−14=¿

o 12+ 13=¿

Extension questions: Where and when did the concept of 0 appear? What are the different bases used by the ancient systems (i.e., we use a base-

10 system, consisting of 10 digits/characters, 0-9) Are there any appearances of those number systems or bases still in use?

Special constantsSeed question:

How did each of the special constants (0, 1, i, e, π) develop?Questions to build on:

Are there any specific people associated with any of the constants? Who and what is their connected?

Are there any similarities among them (properties, computations, graphically, etc.)?

Are there any differences among them? How are they unique?Extension questions:

Are there any other mathematical constants considered “special?” Do any of these constants appear in the natural world? What is the difference between irrational numbers, transcendental numbers,

and normal numbers? Do any of the special constants belong to one or more of these groups?

InversesSeed question:

What is the definition of a mathematical inverse?Questions to build on:

What are some common examples of inverses in computations? How are inverses of functions related graphically? Algebraically? How are the domains and ranges of inverses of functions related? How are the x- and y-intercepts of functions and their inverses related?

Extension questions: Find the inverse of the following functions:

o f ( x )=3 x+7o g ( x )=4 x2−9o h ( x )=e2x

How is function composition related to a function and its inverse? Prove this relationship with an example.

What is a one-to-one function? When is a function considered “non-invertible?” What does that mean? How

can a non-invertible function be made invertible?