Common Core Standards: Math
3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1, 1), (2, 4) and (3, 9), which are not on a straight line.
The artist Paul Klee once said, "A line is a dot that went for a walk." You know what? He was right. A line really is just a collection of points. When a function forms a line, that function is called linear.
Students should already be more or less familiar with linear equations. At least to the point where they get a slight déjà-vu when they see the equation y = mx + b. All you need to do now (aside from refresh their memory) is tell them that this equation for a line is the same as a linear function. As long as x stays with a power of 1, the function will be linear.
Students should be able to recognize linear functions in standard form (like y = 4x + 5 and -½x – 7) and graph them as straight lines. Some points that go on walks keep going on and on in a straight line forever. Other points prefer to travel along the winding scenic route.
Functions that aren't lines are grouped into the nonlinear category. Students should be able to recognize functions that aren't linear. For instance, the equation y = x2 is a function because each x value produces one y value. Yet, it doesn't fit the pattern of y = mx + b. If we graph the function, we end up with a curve called a parabola, shaped like a giant U.
Students should recognize quadratics and absolute value functions as nonlinear. (The graphs of both look like points started going one way and then changed their minds.) While they are still functions (assure students that the definition of "function" hasn't changed!), they are most definitely not lines. And that should be clear from both the graphs and the equations.
If students aren't sure, they can tell whether a function is linear by looking at the equation and seeing whether it fits into y = mx + b. Even an equation like 3x + 9y = 0 can be rearranged to fit the standard form, but y = x2 + 2x + 1 never will. If they still need more convincing, they can plot the points and see whether or not the graph is a straight line.
Whether linear or nonlinear, students should know that equations and graphs aren't always functions. The easiest way to determine whether or not a graph is that of a function is the vertical line test. If a vertical line can be drawn anywhere on the graph without hitting the line or curve more than once, the graph is a function. If not, what we have is not a function, but rather a point with no sense of direction.
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