Scaffolded Math and Science: Graphing algebraic function transformations with cut paper videos

In this post are short videos showing algebraic function transformations using cut paper. a free printable PDF cheat sheet and an explanation as to why "inside is opposite" for horizontal function transformations.

In this post are a bunch of function transformations videos showing how vertex form functions all transform using the same pattern. The way that functions transform in the coordinate plane can feel pretty abstract to algebra 2 students just learning about nonlinear functions. But every algebraic function in vertex form transforms the exact same way.

Functions can translate vertically and horizontally, and even reflect over the x and y axes just like geometric shapes. The one quirk is that horizontal transformations feel opposite from expected.

Why are horizontal transformations opposite? It feels backwards for the vertex of y = |x - 3| to translate right 3 units. With all horizontal shifts, we're looking for the value of x that will "zero out" the inside expression. For x - 3 = 0, x would need to be 3. With x gone, we can find the function's lowest or highest y value, i.e. the vertex's y value.

Here is how we can find the horizontal transformation of a quadratic function in vertex form and why the inside shift is opposite.

function transformations cheat sheet

This function transformations cheat sheet has graphics from an algebra 2 word wall for absolute value, quadratic and square root graphs. There's also a free set of dancing skeleton parent functions posters here.

Below are a bunch of function transformation video shorts using cut paper. In each video you'll see familiar nonlinear algebraic functions transformed in the coordinate plane.

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.

Cubic functions in vertex form are fun to graph. Just a little squiggle through the point.

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.