WOMAN: In this interactive, we explore graphs of parabolas,
parabolas where the equation is given in the form
y = a(x minus h) all squared + k.
This form is referred to as turning point form.
We're currently looking at the graph of y = x squared.
That is, where an equation in turning point form
has an a value of 1, an h value of 0 and a k value of 0.
This interactive allows us
to explore the effects of a, h and k on the parabola.
Let's first consider the effects of a.
a is currently set to 1
and we'll first consider what happens as we increase the value of a.
And we see as a increases, we're getting some sort of a dilation
and, in fact, the dilation that is occurring
is a dilation away from the x-axis.
So the graph we're seeing at the moment is the graph of y = 3x squared
and that's the graph of y = x squared
after a dilation of a factor of 3 from the x-axis.
That is, the point that was at (1, 1) is now at (1, 3).
It's been stretched by a factor of 3 away from the x-axis.
So a certainly affects the dilation of the parabola,
and we note that as we've decreased the value of a
and I've stumbled across a = 0, we're seeing nothing.
Well, in fact, we're seeing the equation y = 0,
which is a horizontal line along the x-axis.
But as we make a negative we see the parabola reappear
and we're now seeing a parabola that is upside down.
So negative values of a reflect the graph in the x-axis.
We're still seeing
as we make the magnitude of a larger, the graph is being dilated,
so the magnitude of a
is a dilation from the x-axis by a factor of a,
and the sine of a
is to do with whether the parabola is up the right way or upside down,
so a negative a value causes a reflection in the x-axis.
Let's now consider the effects of h. h is currently set at 0.
So we'll look at what happens as we increase the value of h,
and we see that as we increase h, the graph is moving to the right.
So here, where h is equal to 6,
that is the equation is (x minus 6) all squared,
the graph has been moved to the right by 6.
Decreasing the value of h,
it would be no surprise to see the graph moves instead to the left.
So here, for example, where h is equal to negative 4,
that is the equation we're seeing is y = (x + 4) all squared,
the graph has been moved to the left by a factor ... by 4.
And so the key thing to note with h
is that when looking at the equation of the graph
if in that bracket you're seeing (x + something) all squared,
the graph has moved to the left by that factor.
If you're, in that bracket, seeing (x minus something) all squared,
the graph has, in fact, moved to the right by that factor,
as is the case here, right by 3.
And lastly, let's consider the effects of k,
and at the moment k is set to 0
and we see that as we increase the value of k,
the graph, in fact, moves up.
So here where k is 8, and hence we have the equation y = x squared + 8,
we've seen the graph move up by 8,
and as we decrease the value of k, making it negative,
we see the graph move down.
So here where the value of k is negative 5,
we're seeing the equation y = x squared minus 5
and the graph, the parabola, has been translated down by 5.
And so, obviously, we could put all of these transformations together,
the dilations and reflections caused by a
and the translations caused by h and k,
to create any kind of parabola in any kind of position.
Now, we mentioned previously that this form of a quadratic equation
is called turning point form
and that's because, as we'll see here, for example,
when we make h 4 and k 5 and hence we have the equation
y = (x minus 4) all squared + 5,
we see a direct correlation between those numbers, those translations,
and the turning point of this graph.
Obviously, this is the graph of y = x squared
after a translation to the right by 4 and up by 5,
and that would therefore make the turning point
have coordinates (4, 5),
and we should be able to read those coordinates
directly out of the equation,
looking at the h value and the k value.
So turning point of a graph in this form
is, in fact, at (h, k).