So, I definitely want to talk about Linear Functions. You may have reached this page because you had trouble answering question 015A of the AMAZEing Math Maze. Or you just wanted to know about Linear Functions. Let's start with the Maze.
Question 015A was about Bob and his figurines. Every month we added 3 figurines to the total number of Bob figurines. If you look at this, you should notice that the number of figurines increases by 3 each month. That is a steady linear rise. The correct answer was Linear. Let's talk about Linear Functions a bit and then ask question 015B.
The key part of of the word Linear is the first 4 letters LINE! These functions form a straight line when plotted on a piece of graph paper. the equation will be in the form y = ax + b where a and b are constant numbers.
Quadratic functions are of the form
when graphed it will form a parabola which looks like this -
Exponentials are of the form
And they look similar to this. Growth or decay starts at a point and then rises (or falls) in ever increasing jumps.
And, while we're at it, let's talk a little about Absolute Values. This is when the portion of the equation within the brackets is positive such as y = | 2x | + 7. The 2x is positive regardless of what value we let x be. So, the graph of an absolute value is mirrored about a line representing where the value of the stuff inside the straight line brackets turns to negative and then back to positive by the brackets. Anyway, for a linear function it looks like this -
Bob is getting better and better at laying bricks. He gets paid by the brick, and each week he makes 2 more dollars than the previous week. It's starting to look like he will keep improving at this rate forever! One could say his skill level increases at a ____________ rate.
For a more complete explanation of Linear Function you may want to check with Columbia University in New York. When I tried to look up the "official" definition, the dictionary type sites referred me to linear transformations. Not very satisfying as linear transformations isn't really what we're looking for here. Any way Columbia University does a pretty good job of explaining things in detail with examples.
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