Algebra 1: Graphing’s a cinch! (For us… maybe not so much for our students)
- We teach linear and quadratic parent functions… and often fail at inverse variation
- We begin to help them understand the usefulness of finding the vertex and substituting values for x.
- We show the connection between evaluating and the graphing calculator table
- We take a stab at transformations, often to little or no avail. It’s difficult for our students to step back and see the larger picture with its common themes.
Algebra 2: Basic transformations of graphs (Okay, we’ve got this!…)
- Function families (quadratic, cubic, square root, cube root, absolute value, exponential, and logarithmic functions, etc.)
- We scratch the surface on horizontal/vertical stretches and shifts, but combining them together is utterly obnoxious.
- We focus on the centrality of the vertex. Pun intended.
Pre-Calc: AHHHHH!! ALL THE TRANSFORMATIONS ALL AT ONCE!
- ’nuff said.
- When they walk in our door, we think they’re set. They’ve seen transformations SO many times! But…. no. I still have IB students that mix up the *words* horizontal and vertical in combination with right/left and up/down.
Then I started taking a table/transformation approach to everything I teach in IB Math SL 1 (Honors Pre-Calc): Trigonometric Equations, Quadratics, Exponents, Logarithms… did I say EVERYTHING?! When you take a few days to explicitly teach students about horizontal and vertical transformations in the moment that it first appears (in my case, trigonometry), the rest of the year is just about reinforcement. Then, you can start focusing on the cool connections between vertical and horizontal stretches/compressions in quadratics. Or the fact that a horizontal shift in an exponential equation is equivalent to a vertical stretch. It blows my mind!
First we discuss each transformation individually, and then we put them all together in standard form. For example, in the equation f(x) = a (b (x – c))^2 + d:
- |a|>1 is a vertical stretch; |a|< 1 is a vertical compression (Although…. once the kids get to IB, the proper phrasing is that EVERYTHING is a stretch. The stretch/compression nature of it has to do with the number that follows. Ex: vertical stretch by a factor of 1/2 versus vertical stretch by a factor of 5.)
- – a is a reflection in the x-axis.
- |b|> 1 is a horizontal compression; |b|< 1 is a horizontal stretch
- – b is a reflection in the y-axis.
- c > 0 represents a horizontal shift right c units, c < 0 represents a horizontal shift left c units.
- d > 0 represents a vertical shift up d units, d < 0 represents a vertical shift down d units.
All that to say: I emphasize the fact that the numbers OUTSIDE the parent function/parentheses affect the y-values, and the numbers INSIDE the parent function/parentheses affect the x-values… and always have the opposite effect of what you’d think.
Once students learn the process of determining graphical transformations, then they have to figure out how to accurately represent those transformations on a graph. *facepalm!* Some teachers:
- (method #1) have students graph each individual transformation until they get to the final result.
- (method #2) have students plug numbers in until they get a general picture.
- (method #3) give up and have students look at the tables in their graphing calculators to come up with exact points.
Don’t get me wrong – there is a time and a place for calculator tables, but I don’t believe that’s where we should begin. I also find that these methods are fairly cumbersome and methods #2/#3 don’t build an understanding of the center and parent function of each graph.
Here’s where the Table Approach is incredibly useful.
1. List the important (x , y)-coordinates for the parent graph in a table of values. For y = x^2, I generally choose these 5 points (but you can always have your students choose more!):
2. Define your transformations from the parent function to the new equation.
- Vertical stretch by a factor of 1/2 (compresses the graph) -> * all y-values by 1/2
- Horizontal shift right 4 units -> x + 4
- Vertical shift down 1 unit -> y – 1
3. Create a new table, with the transformations labeled: x + 4 | 1/2*y – 1
Apply these transformations/calculations to the table of values for the parent function. I recommend that students complete all the transformations to the x-values first and THEN the y-values. It prevents processing overload!
5. Graph the resulting points!
- Vertical reflection in the x-axis
- Horizontal stretch by a factor of 2
- Horizontal translation 1 unit left
- Vertical translation up 3 units
I’ve found that this method is the most effective way to handle multiple transformations of parent functions. However, if you’re not having conversations as to the *whys* of each step, students may fall victim to following a rote process rather than truly digesting it. Make sure that your classes understand that they must perform a stretch/compression BEFORE they translate a graph, because the opposite order would result in an entirely different picture. When students make mistakes in their calculations (and believe me, they will), they should be checking their graphs to make sure the points make *sense*!
Note: I used a quadratic as an example because they’re most common through all levels of math. However, I find that this process is even MORE helpful with exponential and logarithmic functions! Read my next post for more on that.