Rates of Change in Quadratic Functions [AP Precalculus Topic 1.3]
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- Опубликовано: 27 окт 2024
- Quadratic functions have a rate of change that is linear. Meaning the rate of change is changing and it is doing so at a constant rate. This video explains it all!
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Wonderful job in your explanation.
How do you find the constant linear equation that shows you change of rate of change?
This is debated a bit amongst teachers so be careful. I have not seen this specific question come up much but I think there is one question in the AP classroom about it. So first find the change between your points. Lets say the points in a table were (1, 5) (2, 2) (3, 1) and (4, 2). The change in order is -3, -1, 1 which means the change is linear (+2). This tells you the function is quadratic. So we know the slope of our linear change equation is 2. The points are using the x-values and the change. So we can look at points (1, -3) (2, -1) and (3, 1). Again those are the x-values from the original points and the change between the y's as the outputs. So using point slope we need the slope of 2 and any one of those points to produce the equation y = 2x -5. Hope that makes sense. Some teachers would argue that is not the right way to do it, but that seems to be how college board is attacking it. Please let me know if that helped.
@@mporinchak Ohh okay thank you- I was also wondering why you paired the x values with those exact changes- for example, why is it incorrect to say that the the points are (2, -3), (3, -1), and (4, 1)? I basically paired them up starting from the second point (2, 2) and ignored the first point (1, 5) instead of ignoring the last one like your example which was (4, 2).
That's kind of what the debate is amongst teachers, what x values do you pair them up with. But college board seems to use the first.
So quadratic functions have a constant average rate of change, and the changes of those are linear??? Does this mean that the average rate of change of a linear function is constant? I had a question on a quiz about this but I cant exactly remember what it was but it has made me confused.
Linear function have a rate of change that is constant. For example up 3, up 3, up 3. The up 3 is constant. Quadratic functions have a rate of change that changes. But how that change changes is linear. Think of it as the second difference of the outputs that’s linear. The first difference of the outputs shows that the change is changing. The second difference is the change of the change. And that is what is linear.
@@mporinchak ok that makes much more sense, thank you.
How do you get the rate of change on a interval of [0,6]
To find the average rate of change over the interval [0, 6] find the output for x = 0 which is f(0) and find the output for x = 6 which is f(6). Then find the rate of change between the two points by doing (f(6) - f(0)) / (6 - 0). Hope that makes sense.