Roots of Quadratic Equations Studio
We’ve discussed finding the vertex of a parabola. If we
have a quadratic in the form y = a(x – h)^2 + k, then the vertex is at the point
(h,k), indeed the reason for writing the function in the form is exactly that it
lets us spot where the vertex is easily. Now the vertex is interesting because
it is the lowest (or highest) point on the parabola. But sometimes we are
interested in when a quadratic function takes some particular value. In this
studio, we will look at how to find when a quadratic function takes the value 0.
We discussed the quadratic formula in lecture and you can read a justification
of this formula on pp. 172173 of the text. In this lab we will look at patterns
for how zeros are related to the vertex of a parabola. The quadratic formula is
related to these patterns, though we won’t go into details in this lab.
Go to the class web site and download the file
m100qestudio.ods. This file shows the graph of the parabola y = a(x – h)^2 + k
on a fixed window, [10, 10] x [10, 10]. The coefficients a, h, and k, are
adjusted by moving three sliders at the top of the page. The specific values of
a, h, and k are listed to the left of the sliders. The zeros of the function,
that is the x values where the function takes the value 0, or, in geometric
terms, the places where the graph crosses the xaxis, are given below the graph.
Move the sliders around. Observe how changing h and k changes the vertex while
changing a changes how spread out the parabola is.
1. Move the sliders to set the value of a = 1 and h = 0.
Just move the slider for k around. Fill in the missing values in the table
below. Note that the slider for k won’t move far enough to find the answers
for the last row. So how are you supposed to answer that? Look for a pattern
from the answers you find from the first three rows and use that pattern to
deduce what values must go in the last two row.
k 
Zeros 

1 and 1 

2 and 2 

3 and 3 

4 and 4 
25 

2. Now move the sliders for both h and k (while a is still
1). Fill in the missing values in the table below. You will find it helpful to
observe that changing k changes how far apart the two zeros are without changing
the midpoint between the two zeros, while changing h slides the zeros on the
axis, without changing how far apart the zeros are. Once again, you won’t be
able to push the slider far enough to get the last row, but you should be able
to answer the question based on the patterns you find from the first four rows.
H 
k 
Zeros 
Middpoint
Of Zeros 
Distance from
Midpoint to Zeros 


3 and 5 




3 and 1 




4 and 4
(this is called a double root) 




1 and 7 




7 and 3 


3. Describe the pattern for the relationship between the
zeros and the values of h and k. In particular, how are h and k related to the
midpoint of the two zeros and how far apart the zeros are?
4. Solve (x – h)^2 + k = 0 for x in terms of h and k.
Describe how this formula matches the pattern you found for h in the previous
problem. In particular, note where the midpoint of the zeros is according to the
formula and how far apart the zeros are according to the formula.
5. You may have noticed when you moved the sliders around
that sometimes the zeros were listed as Err:502. This means that there were no
zeros, because the parabola never crossed the xaxis. For what values of h and k
are there no zeros? For what values of h and k is there a double root (two equal
zeros)?Does the answer change if you slide a to be 1 instead of 1?
Now that we’ve worked out the easier monic case (with a =
1), we’re ready to tackle the full situation where we can have a take other
values.
6. Now you should move all three sliders. Fill in the
missing values in the table below.
a 
h 
k 
Zeros 
0.5 


1 and 1 
0.5 


2 and 2 
3 


1 and 1 
3 


2 and 2 
2 


1 and 1 
2 


2 and 2 
0.5 


3 and 5 
0.5 


3 and 1 
3 


0 and 2 
3 


1 and 5 
2 


3 and 1 
2 


2 and 6 
(continuation of 6) Solve a(x – h)^2 + k = 0 for x in
terms of a, h and k.
7. Describe how this formula matches the patterns you
found doing problem 6.