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course Mth 173
2/16 4pm
The simulation athttp://phet.colorado.edu/sims/equation-grapher/equation-grapher_en.html
allows you to vary the parameters of a quadratic function and see how the graph changes as a result. You can use sliders to vary a, b and c, or you can enter numbers.
Investigate what happens when you vary the a parameter, and give a brief description of what you see and how it is related to what you are learning in Modeling Project 1.
Varying the a parameter from negative to positive reflects upon the x-axis and as x gets closer to 0, the graph begins to resemble a horizontal line. This relates to slope or `dy/`dt and the rate theat the depth is change with respect to time for modeling project 1. Changing the pressure placed on the column of water or changing the hole’s size would affect steepness of the slope and thus shrink or expand the graph horizontally.
Now see what happens when you vary the b parameter, and once more give a brief description and see if you can relate it to what you have done in Modeling Project 1.
Varying the b parameter affects the vertex, moving left or right depending on the signs of a and b. Modeling Project 1 has a is positive and b is negative so it is shifted to the right.
If you increase the b parameter, does the graph move to the right or the left? Does it move up or down? Does either answer depend on the value of the a parameter?
The answer depends on the value of a. If a is positive and b is positive, then the graph vertex shifts left and down as b increases. If a is positive and b is negative, then the graph shifts right and down as b decreases more and more . If a is negative and b is negative, then the graph shifts left and up as b decreases. If a is negative and b is positive, then the graph shifts right and up so we see that b is related to slope when t = 0. So from the project, we see that it is related to the rate that the depth was changing a starting time = 0 and y = the depth of the water.
If you want to move the vertex to a specific point, can you figure out in advance how much to change the b and c parameters, or are there confounding factors that confuse the process?
Give your best responses, copy them into a Submit Work Form and submit.
We can figure out these values.
We can start out with the basic t^2 function, which in the standard form, the expression is t^2 + 0t + 0 where f(t) = at^2 + bt + c is the function for the depth where a is related to the steepness of our model, b is related to the slope when t = 0 and c is related to the y intercept. If you wanted to move the vertex to (1, -1) with a = 1, then we would have f(t) = at2 - 2aht + (ah2 + k) where (h, k) is the desired point. So f(t) = 1t2 - 2*1*1t + (1*12 - 1) = t^2 - 2t + 0 thus b = -2 and c = 0.
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Excellent analysis throughout.
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