Assignment 2 #4: Interpret your graphs. What happens to y = ax^2 + x +2 as a is varied? Is there a common point to all graphs? What is it? What is the significance of the graph where a = 0? Do similar interpretations for other sets of graphs. How does the shape change? How does the position change?

The graph of y = ax^2 + x + 2. I will
find out if there is a common point to all the graphs if a is
varied in this problem. As you can see, this equation graphs a
parabola and it intersects the y - axis at positive 2. My hypothesis
#1: as **a** changes, the parabola will widen; hypothesis
#2: but, it will continue to cross the y - axis at positive two.
I will continue to find out if my hypothesis holds true. Keep
in mind that in the equation, ax^2 + x + 2, **a** is a coefficient
of x^2, the coefficient of x is 1, and the constant is 2.

The graph of 2x^2 + x +2 and the graph of ax^2 + x + 2. My hypothesis #1 was incorrect, however, my hypothesis #2 was correct. What will happen if I vary a to a negative number? Will it widen the parabola? Will it change the parabola in the opposite direction? Will it make the parabola become a hyperbola? You must be able to distinguish between the two terms!

As you might have guessed, the parabola is flipped to the other side of the graph and still intersects the y-axis at positive two.

Let's take a look at the graphs of positive,
even integers. Look at what the graph shows when we set** a**=
2, 4,6,8,10,20, and 100.

What type of conjecture can you make
about these types of equations? Notice that all parabolas cross
the y-axis at positive two. What can we say about this? What will
happen when we set **a** = odd numbers? What will happen when **a**
= 0?

The graph of ax^2 + x + 2, where a =
1, 3,5,7,13, and 101. There is no significant difference between
the graphs of evens versus odds, but if you notice how wide the
parabola is on the **yellow** curve (this is the graph of x^2 + x + 2) versus
the graph of the red curve (this is the graph of 101x^2 + x +2).
Still all graphs are intersecting at y = 2.

We will now turn our attention to the
graph of ax^2 + x + 2 where **a** = 0. What do you think will happen to our graph?
Keep in mind that when you multiply zero by any number, your product
will be....you guessed it...zero!

I hope you guessed a straight line as I did, because if you have x + 2 left, this equation automatically results in a linear function. This equation crosses the x and y axis at x = -2 and y = 2.

What do you think happened to this graph?
What types of values were the inputs? All graphs again cross at
y = 2 except for the **yellow** graph. Why is that? All constant values in the
equations above stayed the same except for the **yellow** graph,
therefore the constant changed in the equation, to 10 (this is
where the parabola crosses the y-axis. The **light
blue** graph seems to "protrude"
in this graph and the reason is due in part because I gave a coefficient
to the x value, so the equation became y = 5x^2 + 5x +2. The **purple** graph
is that of the equation y = (1/2)x^2 + x+2. It seems to look a
lot like the graphs of the equations above. The **green** graph
is that of y = 0.14114111411114x^2 + x+2 (the coefficient of x^2
is an irrational number. The graph of this equation is very wide,
but it does still cross through the y-axis at y = 2. You have
learned what the **red** line's equation is, so no further discussion
is needed. I wonder if you can recall the equation of the **blue** graph (or
one that is similar to it)? If you said it was a negative graph,
you are correct!!! In fact it is -(1/2)x^2 +x+2.