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Graphing the Transformation y = a f(x) + k - Problem 1

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

We’re graphing transformations. Here’s an example; let’s graph y equals -4 times root x plus 8. First thing you want to do is identify the parent graph that’s being transformed. It’s this function f(x) equals the square root of x, and it helps if you remember what the shape of that graph is. It looks kind of like this. It’s got key points (0, 0) and (1, 1).

Now I’ll mark those points down here because my method for graphing these functions, these transformations, is usually to start with a parent graph, transform the points and then plot the transformed points. What do I have to do to these points? Well this formula tells me I have -4 times root x plus 8. I’ve got to multiply the square root of x times -4. These are the square root of x. Multiply them by -4 and add 8.

The x values I can just leave as they are, so 0 and 1. I take this rot x value, multiply by -4, 0 times -4 is 0 and then add 8 and I get 8. I take this value, multiply by -4, I get -4 and add 8 and I get 4. Let’s try another value. It pays to use perfect squares when you’re dealing with the square root function so I’m going to use 4 and the square root of 4 is 2. Now the x value is going to be exactly the same over here but what happens to the 2? We multiply by -4 and get -8 plus 8 is 0.

This gives me three points to work with, (0, 8), (1, 4) and (4, 0). It’s really important to remember, (0, 0) this point was really important it was the end point of the graph. And that point has become (0, 8). So (0, 8) is right here. Then we have (1, 4) that’s 2, that’s 1, (1, 4) would be about here. And then we have (4, 0) right here. Our graph is going to look like this. It’s a downward opening square root graph, and here’s the end point at (0, 8) and that’s our final graph.

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