# Second Degree Equations

Ronald Aguilar

Here we investigate the equation and we are looking at the variable a which is the coefficient. First, lets graph the basic equation for a parabola, .

Notice how the graph looks like a U. Now that we have seen the graph for , lets see what happens when we make 0 < a < 1 and a < 1. Here we use (graph in blue) and     (in red).

Notice how the graph  of   gets wider as compared to .    If we plug in 1 for x the outcome of the function is 1/2 (y = 1/2).   Since the function is even if we plug in -1 for x the outcome is the same, 1/2 (y = 1/2). The graph   grows slower than  . The graph of  gets slimmer as compared to . If we plug in 1 for x the outcome of the function is y = 2. Since the function is even if we plug in -1 for x the outcome is the same, y = 2. The graph grows faster than .

After exploration of the positive coefficients of a parabola, the positive quadratic , the positivity of the parabola reminds me of an U. This U reminds me of a smile. Lets look at the negative coefficients of the equation.

The negative quadratic is shown below where a = -1.

This is the graph upside down, reflected across the x axis. Let's figure out why. Using the fucntion , we plug in 1 for x, we get y = -1. If we look at the graph we can see that graph intersects at (1,-1). Moreover, this function is odd. Plugging in -1 for x, gives us y = -1. The graph intersects (-1,-1) too.

Change the values of a gives us a better idea for how the value of a affects the graph. Below are the graphs of    (in black) and (in red). As we can see the graph of is wider than the graph . The graph of is slimmer than the graph of . If we plug in 1 for x in both equations as before, we can see why the graphs behave as they do.

The negativity of the parabola reminds of an upside U. This upside down U reminds of a frown.

Lets explore the different variations of the coefficient a in the parabolic equation .

The graph of is in purple. The negative of this equation is . This graph is shown in red. This graph is reflected across the x axis. If we plot the points like we did before we can see why these graphs reflect each other across the x axis. The same goes for each graph here.

Changing the value a indicates how fat/thick/wide or skinny/thin/slim the parabola will be.

1)If the absolute value of a () is less than 1 then the parabola gets wider because the equation grows slower.

2) If the absolute value of a () is greater than 1 then the parabola gets slimmer because the equation grows faster.

So in all actuality the coefficient determines how fast the parabola grows.