Unit
Quadratic Equations and Inequalities
University of Michigan
Runs his own tutoring company
Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!
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University of Michigan
Runs his own tutoring company
Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!
We are used to looking at quadratic equations where "y" is the variable that is equal to the squared "x" terms. However, in a horizontal parabola the "x" is equal to the "y" term squared. Instead of going up and down, a horizontal parabola goes from side to side. When graphing a horizontal parabola, we first need to make sure the formula is in standard form and then plot accordingly.
We are used to looking at quadratic equations
that are Y is equal to something squared.
Okay.
But that doesn't always
have to be the case.
We could end up dealing with X is
equal to something Y squared.
And what that really does is take a parabola,
which we are used to looking at
as up and down, so either straight, coefficient
of positive A or negative A. And what it does instead is
make it a sideways problem.
So you're dealing with something like
this or something like this.
Okay.
Trick is that this isn't
actually a function.
Remember that in order to have a function
we have to have one Y for every X.
So if you look at something like
this, that's not going to hold.
So this isn't a function, but that doesn't
mean we can't still try to graph
it.
So what we can do is in order to graph it,
think about what the relating equation
is for Y equals.
So I have right here X is equal to Y
plus 1 squared minus 3. Think about
the relating graph, Y is equal to
X plus 1 squared minus 3. And
think about what each component does.
The minus 3 moves the graph down three.
So that is taking the Y value down three.
If we are dealing with the X equals,
it's going to do the same thing.
Instead of taking the Y down 3 it takes
the X down 3. So what you're doing
is making the X three units smaller.
So we would take our vertex and move
it back 3. Similarly to the X plus 1.
What this is going to do is move our X
coordinate of our vertex back one for
a vertical parabola.
It's going to move the Y back one
on a side-to-side parabola.
By taking the Y back one what that means
is making the Y one smaller or moving
it down one.
We don't have any coefficients to make
this any steeper or wider, so what we
found out is that our vertex is back
three, one, two, three and down one.
Our coefficient is positive, which tells
us we have in this case an upward facing
parabola where our Y values are
going to be getting bigger.
Here we have no coefficient which then tells
us the opposite, our X coordinates
are going to be getting bigger.
We're dealing with negative Xs and
it's going to be going this way.
We're going to have something like this.
Again, you could always plug in a point
if you want to get a little bit more
precise graph.
Here we're dealing with negative one.
So I would probably end up plugging in
Y equals 0 to get an extra point.
But what we've done is basically graphed
a horizontal parabola by finding the
vertex, pretty much the exact same way
as we did before but just switching
our X and our Ys.
Just to note, again this is not a function.
The two pieces together, we could break
it up and look at either just the top
or just the bottom and those are functions.
But together when we're dealing with an
entire sideways parabola, it's not a
function.