More on Parabolas
By: Ginger Rhodes
Look at y = x 2 + bx + 1 for different values of b. What do you notice about the “movement” of the parabola as b is changed? Click HERE for an animation.
Notice the y-intercept is the same point (0, 1) for each graph, which makes sense because if you substitute zero in for x you will always have the + 1 to get an y-coordinate of 1.
At a closer look there are two roots when 2 < b and b < -2,
one root when b = 2 and b = -2, and no roots when -2 < b < 2.
Now, what does the graph of the locus of the
vertices of the set of parabolas graphed
when b varies in y = x 2 + bx + 1 look like?
Remember, the vertex can be found using the
line of symmetry x = -b / 2a.
First, find –b / 2a (the x-coordinate of the vertex)
and then substitute in for x to find the y-coordinate.
Since a = 1 we can use –b/2.
I will substitute x = -b / 2 into the following equation
y = x 2 + bx + 1
y = (-b/2)2 + b(-b/2) + 1
So, y = -b2 / 4 + 1
Therefore, the vertices include
(-1.5, -1.25), (-1,0), (-.5, .75), (0,1), (.5, .75), (1,0), and (1.5,1.25).
It appears to be a parabola!
How can we find the equation of the parabola?
We know x = -b/2 and y = -b2 / 4 + 1,
so let’s solve for b and set the two equations equal to each other.
What happens as c changes? Click HERE to explore.
What happens as a changes? Click HERE to explore.
Let’s take a different look at the equation 0 = x 2 + bx + 1.
What does it look like in the xb plane?
Now, suppose we take b = 3, and overlay this equation on the graph. Notice we add a line parallel to the x-axis. Where the horizontal line b = 3 intersects the graph is the roots for the original equation.
Notice, at a closer look there are two roots when 2 < b and b < -2,
one root when b = 2 and b = -2, and no roots when -2 < b < 2 (just as before!!)
What happens when c = -1?
Other c values?
c = -3, -2, -1, 1, 2, and 3
What about c = 0?
What relationship does 2x + b = 0 have with the other graphs?
It appears to go through the vertex.
Thinking back to before we know this point is a single root and where the quadratic is tangent to the x-axis.
This point is also the vertex.
Before we discussed x = -b / 2a is the x-coordinate of the vertex.
Solve for b and substitute a = 1 you get b = -2x.