Algebraic function graphs and their transformations can be challenging at first. Why graphs shift in the coordinate plane the way they do can feel pretty abstract. Vertical transformations shift as expected, but horizontal shifts are "opposite"-- why? With practice, students start to "get it" and can even apply what they know about one function graph to an entirely new function. Our function transformations unit was always one of my favorite units to teach in algebra 2 because of this.
Over the past week or so, I've made 7 function transformation video shorts using cut paper. In each video you'll see familiar algebraic functions—like quadratic, absolute value, and radical functions—transformed visually in the coordinate plane. There's also one for sine.
We started our function graphs unit every year by transforming absolute value graphs. There's an absolute value cheat sheet for graphing linked in this post. First we find and plot the vertex, then choose x values on either side of the vertex's x, then calculate all y values by evaluating the equation at our x values.
Next up was graphing quadratic functions in vertex form. Just like absolute value, we created tables centered on the vertex to plot all points.
Graphing quadratics in factored form always came a little later, but I thought it would be fun to include a video for this, too. When we get to factored form quadratics, the "horizontal shifts are opposite" starts to make sense.
What x values would make (x - 5)(x + 7) = 0 true? +5 and -7. All horizontal function shifts are just like this. We're looking for x values that will zero out the parenthesis, which is why "inside is opposite".
After quadratics, we'd move on to radicals. By now, function transformations are really clicking with students, and they are able to move a graph all around the coordinate plane. Most of my years teaching algebra 2 have been spent in the special education setting. It's cool seeing students who can be unsure of their math abilities confidently shifting radical functions by the time we get to this unit! There's a free graphing radicals cheat sheet linked in this post.
Towards the spring, we'd learn how to sketch polynomial functions. This was another unit in algebra 2 where students showed a lot of growth. At first, graphing a polynomial seems completely abstract, but soon enough students are crossing and bouncing their graphs on the x axis and sketching in their graphs! There's a free sketching polynomials cheat sheet, along with a couple activities in this polynomials post.
After polynomials, we'd move on to graphing exponential functions. There's a free graphing exponential functions cheat sheet linked at the bottom of this post.
We'd get to graphing logarithms towards the end of the year in our inclusion class. There's a free graphing logs cheat sheet linked in this post.
At the very end of the year, our inclusion students would complete a conics project. It was so inspiring seeing what they created with their mathematical equations. If you've never done a conics project, there are examples and tutorial videos in this post.
Lastly there's sine, which isn’t an algebraic function but I wanted show how it shifts in the same way. My original graph looked like a cubic, and I rightfully heard about it from a couple people on Facebook. So I deleted the video, beat myself up for a couple days, remade the graph, and then remade the video.
I hope you enjoyed the videos! Is there another function you’d like to see?
> Browse all function activities.
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