Compare the graphs of
quadratic functions in the form: y = ax2 + bx + c.
a. What are the coordinates of the y-intercept?
b. What must be true in order for the graph to reflect over the y-axis?
c. What must be true in order for the graph to open down? to open up?
What must be true in order for the graph to touch the x-axis only one
time? Zero times?
e. How can you calculate the equation of the axis of symmetry (the vertical
line where the graph can be folded in half)?
f. Where on the graph is the solution for ax2 + bx + c = 0 ?
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This exploration wants me to
determine what happens to the graph when certain parameters of a Quadratic
Solve/Investigate the Problem
I used my handy dandy TI
84 Silver Edition (can you tell how proud I am of the calculator?) to
determine what would happen. I
entered the following Quadratic Equations
Below, you will find a graph of the
functions. You will have to
imagine that the Y axis appears on the gridline that is marked 0 along the
Notice that when my constant
term changes, my graph is shifted up the number of positive units and down
for the number of negative units.
Part B What must be true in order for
the graph to reflect over the y-axis?
Here, if you look at the
equations used, the only difference is in the sign of the second term. Now, IMAGINE that the Y axis is at
the 0 gridline. You will
hopefully observe that the curves are a reflection of each other. So it appears that changing the sign
of the middle term will reflect your parabola across the Y Axis! HOW EXCITING!!
Part C What must be true in order for
the graph to open down? to open up?
Take a look at the Equation
Box. How do the equations
compare? The only difference in the two is the sign on the LEADING
So, it appears that the sign of
the leading coefficient determines the direction of the parabola!
Part D. What must be true in order for
the graph to touch the x-axis only one time? Zero times?
For this, I will refer to my pre existing knowledge
and you can take a look at the link, to get a quick lesson on FACTORING.
When you factor a QUADRATIC EQUATION, you can go one
step further to determine the
X Intercepts. (X intercepts are the places where
the graph crosses the X Axis)
Look at the following factoring thingy! (Give me a
break! You KNOW what a THINGY
The only time that the Parabola
will just touch the axis once is when you are dealing with a PERFECT SQUARE
TRINOMIAL (PST). In the
Factoring Thingy, you should observe that the factored form has 2 identical
factors. Recall that when you
multiply a thing by itself, you are really SQUARING that thing. Since (x+9) (x+9) = (x+9)2,
the original equation is considered a PST. The root (another term for the x
intercepts) is actually a double root since it appears twice when
solving. Hence, there is only
You will have to forgive my
laziness. I just really
don’t feel like explaining the line of symmetry thing. Take a look at this link, and
amuse yourself! The equation that goes through the vertex,
is the line of symmetry.
Part F Where on the graph is the
solution for ax2 + bx + c = 0 ?
If you read carefully, you will see that I have
already answered this question!
Factoring and using the ZERO PRODUCT PROPERTY yields the Zeros
– Roots – X intercepts of a Quadratic. If a Quadratic is not factorable,
utilize the QUADRATIC
of the Problem
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