High School: Functions

Building Functions HSF-BF.B.3

3. Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx) and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Much like replacing sugar with salt can transform a blueberry pie to a foul-tasting disaster, replacing x with x + k or kx can transform a graph. Students should know that adding a constant k to a function will change the graph of the function depending not only on the value of the constant, but on where it is inserted as well.

If y = f(x) is changed to y = f(x) + k, the curve will shift vertically (up for k > 0, down if k < 0). Adding k to x such that y = f(x + k) will shift the curve horizontally (left for k > 0, right for k < 0).

Multiplying f(x) by a constant k stretches (k > 1) or squishes (0 < k < 1) the graph vertically. If k < 0, the graph is also flipped over the x-axis. Multiplying x by k stretches (k > 0) or squishes (k < 0) the graph horizontally.

Students should also know that by definition, a function is even if f(-x) = f(x). If students are confused as to how this happens, give them the function f(x) = x2. It's even because f(-x) = (-x)= (-1)2 × (x)= 1 × x2 = xf(x). Make sure they know not all functions with even numbers are even functions! It's an unfortunate and too common mistake. Even functions are symmetrical across the y-axis.

An odd function is a misnomer because plenty of odd functions aren't strange in the slightest. A function is odd if f(-x) = -f(x). One such function is f(x) = x3, because f(-x) = (-x)3 = (-1)3 × x3 = -1 × x3 = -x3 = -f(x). Convinced? Odd functions are symmetrical about the origin, not across any axis.

The best part is that students can have fun squishing and moving and flipping curves to their hearts' content without any nauseating repercussions. That's more than we can say for that pukeberry pie.