### Melissa Silverman

Continuing my thoughts from Assignment 1, it is important for students to isolate parts of equations and manipulate them so that they can determine the role of that component in the equation. For this assignment, I will explore using the equation

Before I begin manipulating the equation, I will graph the equation when a=1 to begin my understanding of the equation.

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From my intial graph, I can see that the basic form I am working with is a parabola. I can begin to make a few hypotheses about parabolas from my initial graph. I chose a to be a positive number and the parabola faces upwards. What if a were negative? Would it point down? To test this hypothesis, I can set a=-1.

The vertex, or center point, of the parabola has not changed. The curvature of the parabola has not changed. The parabola now opens downward. So by experimenting with the sign of the coefficient, one may conclude that a positive coefficient will result in an upward facing parabola while a negative coefficient will result in a downward facing parabola.

A conclusion has been drawn about the parabola in regards to the sign of the coefficient. What about the value of the coefficient? As the coefficient becomes larger or smaller, how does that affect the shape of the parabola? By plugging in various coefficient values and graphing them on the same graph, I can draw some conclusions.

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The center, or vertex, of the parabola remains at the origin. All of the chosen coefficients are positive, and all of the parabolas are facing upwards. The only difference in the graphs is the width, so the conclusion can be made that the larger the coefficient, the narrower the parabola. Inversely, the smaller the coefficient, the wider the parabola.

Neither the sign of the coefficient nor the value of the coefficient change the parabola's center. What if I want to move the parabola to the left or right? What can I change to move the vertex along the x-axis? I will try adding and subtracting a constant to x before I square the term.

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Subtracing a constant from the coefficient (before squaring) moves the vertex along the positive x-axis. The vertex's movement corresponds exactly to the value of the constant. For example, the blue parabola has moved over 2 units, and 2 is its constant. I can move the parabola's vertex to any positive x value by subtracting that value from x. I can now infer that adding a constant to the coefficient will move the vertex along the negative x-axis.

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As predicted, adding a coefficient moves the vertex along the negative x-axis. The vertex's movement corresponds to the value of the constant.

The last manipulation is moving the parabola along the y-axis. To accomplish this, add or subtract a coefficient after the squared term.

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Adding or subtracting a constant after the squared term lowers or raises the vertex along the y-axis, respectively. Furthermore, the value that I add or subtract from the parabola is the parabola's y-intercepts. Thus I can control where the parabola crosses the y-axis.

By graphing the equation

while manipulating it in different ways, we have the ability to produce desired parabolas and make predictions about the graphs of equations in this form. By changing the coefficient and inserting coefficients in different combinations, one can control the orientation, width, and vertex placement of a parabola.

This activity fosters flexibility to students in their understanding of parabolas. They can look at parabolic equations and predict what the graph will look like. They can also look at a parabola and create the corresponding equation. Knowing how each part of the formula works means they can take any graph or equation of a parabola and manipulate it to meet their needs. Just as students need the flexibility to think in decimals and fractions, students need to be able to think of equations in terms of their graphs and graphs in terms of their equations to reach higher mathematical understanding.

To continue my thoughts from the previous exercise, this activity fosters curiosity and a list of questions. I formatted my assignment the way a student might think. Beginning with a simple parabola, what questions might I have about it? My line of questioning focused on ways to get the parabola I wanted by changing the formula. By focusing on things that I could change about the graph, I discovered how to determine whether the parabola faces up or down, how to move the vertex of the parabola, how to affect the curvature of the parabola, and even how to determine the y-intercept. By plotting several examples on one graph, I not only determined what was being changed but how it was being changed. More specifically, how does a coefficient or constant's value affect the parabola? These are all logical steps that students could follow and have much sucess with. Allowing students to discover mathematics on their own is sometimes hard to accomplish. Technology allows students to tinker and experiment, learning on their own what would traditionally be presented through textbook pictures and class notes.