Why don't my students understand translations?

Translations are a topic that students struggle with. Why?  Because they can't visualize them. They don't know what the functions look like or how adding/subtracting numbers affects the parent function. 

First, I would launch by using Desmos. Have students graph y=x^2 then y=x^2 + 1, y = x^2 + 2 , y = x^2 + 3 and look for the pattern.  (Great application of looking for repeated reasoning!)

Next, have them graph y = x^2, then y=x^2 -1, y = x^2 - 5, etc and look for the pattern. 

Then, y=x^2 with y=(x-1)^2, y=(x - 4)^2, etc. Then y=(x+3)^2, y=(x+5)^2, etc and look for patterns. 

Once they see the shifts, (up, down, left, right). Check it with something concrete.  

I used a sticky wicky that I found in my house and a small sheet of graph paper. You could give each student a pipe cleaner, they can bend them to form a quadratic then shift them on the graph paper. You can give them different equation to graph and they can physically move the pipe cleaner on their graph paper to where they would shift from the parent graph. 

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