Write-up 2
When beginning to work with quadratic equations, we usually
start with quadratic equations in the form of 
explore
with different values of a. In the following examples
we will look at positive and negative integer values for a and positive
and negative rational values for a.
Here we will look at the following equations in order to come up with a hypothesis.
The above picture shows the graphs when a is a positive
or negative integer. The yellow graph that opens upward is the equation
. The rest of the graphs that open upwards are examples
of what happens when we increase the value of a. Here we let a= 1,2,3, and
4.
The green graph that opens downward is 
.
The rest of the graphs are examples of what happens when we
decrease the value of a. Here we let a= -1,-2,-3, and -4. The graph of 
is between the light blue and the dark gray. It is such a
light gray that on here it looks like it is not there.
From the above we can already make some assumptions. First of all, when a is positive the graph is going to open upward and when the graph is negative it will open downward. Secondly, as we either increase a to be an integer greater than one the graph begins to close up or as I tell my students it is trying to eat the y-axis. Keeping that in mind let us now look at examples when a is a rational number.
I will now look at examples when a is a rational number.
Some examples we will look at is when a= 
, 
,
, and 
.
The graphs below are examples when a is
equal to the above mentioned values for positive and negative values.
As we can see from the above graphs the opposite occurs when a is a rational number. We can see that the graphs are "pushing" themselves down on the x-axis.
1.Therefore, from are findings we know that when a is a value that is larger than 1, our graph becomes steeper.
2.When the value of a is between 0 and 1, the graph is wider and therefore flattens out towards the x-axis.
3. A conclusion we can make from 1 and 2 is that how steep or wide our graph is depends on the size of a. Whether a is an integer or rational number does not necessarily determine the steepness or wideness of our graph.
4.Whether a is positive or negative determines what way our graph will open. If a is negative then the graph will open downward. If a is positive, the graph will open upward.