Teachers frequently get questions from students who are very astute or just curious. A math question that seems simple may have a difficult answer or the answer may connect in unexpected ways to other ideas. This text aims to address commonly asked questions about the algebra curriculum that require expert knowledge to answer.

Questions addressed:

• I don't understand the "machine" analogy for functions.
• Why does a function have to pass the vertical line test?
• How is a sequence different from a function?
• Why are there different notations for functions?
• Why do we switch x and y when finding an inverse function?
• Why does infinity always have a parenthesis in interval notation?
• Why do some absolute value inequalities use AND while others use OR?
• Why should I care about polynomials?
• Factoring quadratic polynomials seems like magic.
• Is it worth learning the process of completing the square?
• Why do we get a system of linear equations from partial fraction decomposition?
• Why does polynomial division work?
• Is factoring polynomials like factoring integers?
• Why do we learn about conic sections?
• Can you explain how multiplicity of roots affects the shape of a polynomial's graph?
• Is cosine an x coordinate or opposite over adjacent?
• How are trig functions related to exponential functions?
• What is the value of cos(i)?
• Why do we learn radians?
• Why do we learn trig identities?
• Why is probability always between 0 and 1?
• Why is the probability P(A or B) sometimes P(A)+P(B) and sometimes not?
• Why is the probability P(A and B) sometimes P(A)*P(B) and sometimes not?
• Why do we learn about the binomial theorem in the probability chapter?
• What is the difference between an average and a weighted average?
• What does a negative or fractional exponent mean?
• Why is (-1)^3 defined, but (-1)^pi is not?
• Why are we not allowed to write a radical on the bottom of a fraction?
• What is so special about e?
• Why are there three different forms for a line equation?
• What is a dot product?
• What is a determinant?
• What is a cross product?
• Why do we learn both substitution and elimination?
• What does it mean for two lines to be parallel in 3D?
• Why is the slope of the perpendicular line -1/m?
• Why is sine an odd function and cosine an even function?
• Why does an odd function times an odd function result in an even function?
• Why does (x-3)^2 move the graph right and x^2-3 move the graph down?
• Why do trig functions have both a vertical and horizontal dilation whereas most other functions have one or the other?

If you have a burning question that requires a technical answer, leave it as a comment and I may consider writing an answer to it in an update.