Hello. I am a homeschooling mom using your DVDs and the Jacobs text. I also happen to have a bachelors in math and taught Algebra in VA for 5 years. (But that was 14 years ago now.) In the Set I questions we are sometimes asked review type questions about functions. Sometimes we are asked “What type of function is this?” How do we know when to answer “direct variation” and when to answer “linear function.” I do know the difference between them, but not sure when a linear function IS a direct variation, which answer is correct. Hope my question is not too confusing…Thanks so much!
Direct Variation equations are also linear equations, so understandably they get overlapped. The distinction is the y-intercept. Direct Variation lines will always go through the origin Their y-intercept will always be 0, that’s why their equations always fit the form of y = ax. It is the same as a line which you have y = mx + b, but the b term is 0 for direct variation, so it shortens to y = mx (or y = ax, which is the same thing).
When you are answering the questions in the book, a lot depends on what you are studying in the text, or that section as to whether the book is looking for direct variation or linear. The distinction, though, is whether or not it goes through the origin. If you have an equation that is both a linear function, but also a direct variation equation, I would allow both answers as correct. If you want to be technical about what’s correct, then the answer to “what kind of function is this” would probably be “linear function with a direct variation equation” since direct variation is a special case of linear function. But take into context what the student is learning. If they have already learned what linear equations are, and are now in a lesson specifically designed to teach them about direct variation, then when they come across a problem that graphs as a line through the origin, then the likely answer is “direct variation”.
Does that help?