Welcome to a lesson on transformations
of quadratic functions.
Let's begin by reviewing the graph
of the basic quadratic function
f of x equals x squared.
Remember we can always
make a table of values
in order to graph a function.
Where the x values are the inputs,
and the corresponding
function values are y values
are the outputs.
Each ordered pair is in the form x comma y
or input comma output.
Each ordered pair represents a point
on the graph of the function
as shown here on the right.
Let's begin investigating transformations
of the basic quadratic function.
Let's first look at a quadratic function
in the form y equals x squared plus k
or f of x equals x squared plus k.
Let's see how the value of k affects
the graph of the basic quadratic function
shown here on the right.
Notice when k is positive, the
graph is shifted up k units.
And when k is negative,
the graph is shifted down
the absolute value of k units.
Let's summarize these findings.
The graph of f of x
equals x squared plus k
shifts the graph of f of x
equals x squared vertically.
Where if k is positive
or greater than zero,
the graph is shifted up k units.
Looking at the graph on the left,
notice in green, we have the graph
of f of x equals x squared plus three
which is the graph of the
basic quadratic function
f of x equals x squared
shifted up three units.
If k is less than zero or negative,
the graph is shifted down the
absolute value of k units.
In red, we have the graph
f of x equals x squared minus three.
Notice how k is negative three.
And this graph is the graph
f of x equals x squared
shifted down three units.
Now let's consider the quadratic function
in the form y equals the quantity
x minus h squared or f of x equals
the quantity x minus h squared.
Notice when h is positive, we would have
minus a positive in the parentheses,
and when h is negative, we
would have minus a negative,
which is equivalent to adding a positive
inside the parentheses.
So when h is positive, notice
how the graph is shifted
to the right h units.
And when h is negative, the
graph is shifted to the left
the absolute value of h units.
But again, when h is negative,
for example, here, where
h is negative seven,
inside the parentheses,
we would actually have
x minus negative seven
which is equivalent to x plus seven.
So if we have addition
inside the parentheses,
the graph is shifted to the left,
and when h is positive, for example, here,
where h is positive five,
inside the parentheses, we
would have x minus five.
So if we have subtraction
in the parentheses,
we're going to have to
shift it to the right.
Let's summarize these findings.
The graph of f of x equals
the quantity x minus h squared
shifts the graph of f of x
equals x squared horizontally.
If h is greater than zero or positive,
the graph is shifted to the right h units.
Looking at the graph in red,
here we have the function, f of x equals
the quantity x minus three squared.
Because the subtraction
is part of the form,
here h positive three,
and the graph is shifted
right three units.
If h is less than zero or negative,
the graph is shifted to the left
the absolute value of h units.
So in green we have the function f of x
equals the quantity x plus three squared.
But because the form is this form here,
where we have subtraction,
we need to think of this
as x minus negative three,
so because h is negative three,
the graph is shifted left three units.
Just remember if we have
subtraction in the parentheses,
the graph is shifted to the right.
If we have addition
inside the parentheses,
the graph is shifted to the left.
Now, let's look at the quadratic function
in the form y equals ax squared,
or f of x equals ax squared.
Let's first only focus
on positive values of a.
Notice right now a is equal to one
which gives us the graph of
the basic quadratic function
f of x equals x squared as
shown here on the right.
Let's see what happens
when a is greater than one.
Notice how when a is greater than one,
the graph becomes narrower,
or we could say the graph
is vertically stretched.
Now let's see what happens to the graph
when a is between zero and one.
Notice that when a is
between zero and one,
the graph becomes wider,
or we could say the graph
is vertically compressed.
So again, if a is greater than one,
the graph is narrower,
or vertically stretched.
And if a is between zero and one,
the graph is wider, or we can
say vertically compressed.
Now see what happens when a is negative.
Notice if a is negative,
all of the y values
are going to be negative.
So when a is negative,
the graph is reflected
across the horizontal axis or x-axis.
Let's summarize these findings.
Let's first summarize the
findings when a is positive
or greater than zero, the
graph f of x equals ax squared
is wider or narrower than the
graph f of x equals x squared.
If a is between zero and one,
the graph is wider or
vertically compressed.
Looking at the graph on the left,
in green we have the graph
f of x equals 1/2 x squared,
or notice a is 1/2, which
is between zero and one,
and the graph is wider
or vertically compressed.
If a is greater than one,
the graph is narrower
or vertically stretched.
In red, we have the graph
f of x equals two x squared
where a is two and therefore,
the graph is vertically
stretched or narrower
than the graph of the
basic quadratic function
f of x equals x squared.
To graph a quadratic
function with a constant a,
it's easiest to choose a few points
on the graph of the
basic quadratic function
and then multiply the y values by a.
When we have a quadratic
function in the form
f of x equals ax squared,
and a is less than zero of negative,
the graph is reflected
across the horizontal axis
or x-axis as shown here.
Where in blue we have the
graph f of x equals x squared,
in green, we have f of x
equals negative 1/2 x squared.
So the graph is reflected
across the horizontal axis
and it's wider than the basic graph.
In red we have the graph f of x equals
negative two x squared,
where because a is negative two,
the graph of the basic
function is reflected
across the horizontal
axis and it's narrower.
And for f of x equals negative x squared,
we just have the basic
quadratic function reflected
across the horizontal axis.
Now let's put these three forms together.
Let's look at a quadratic function
in the form f of x equals a
times the quantity x
minus h squared plus k.
More specifically, let's
graph f of x equals two
times the quantity x minus
one squared plus three.
We notice a is equal to two,
h is equal to positive one,
and k is equal to positive three.
If we wanted to graph this function,
we could always make a table of values.
But let's practice graphing it
using what we learned
about transformations.
We will graph the given function in steps.
We will first graph f of
x equals two x squared
where a is equal to two.
Then we'll graph f of x equals two
times the quantity x minus one squared,
where a is two and h is one.
And then finally we'll
graph f of x equals two
times the quantity x minus
one squared plus three,
where a is two, h is one, and k is three.
This will give us our final graph.
So to begin to graph f of
x equals two x squared,
we multiply the y values
of the basic quadratic function by two.
So looking at this graph,
the graph of f of x
equals x squared is graphed here in black.
To graph f of x equals two x squared,
we select several points
on the black graph,
multiply the y-coordinates by two,
and plot the new ordered pairs,
which will give us the blue graph
which is a graph of f of
x equals two x squared.
Notice how this graph is narrower
or vertically stretched compared
to the basic quadratic
function graphed in black.
Now that we have the graph of
f of x equals two x squared,
we will graph f of x equals two
times the quantity x minus one squared.
The graph of this function is the graph
of f of x equals two x squared
shifted right one unit,
because of the minus one
in the parentheses, or
because h is positive one.
So looking at this graph,
again, the blue graph is f
of x equals two x squared,
if we shift this graph right one unit,
the green graph is the
graph of f of x equals two
times the quantity x minus one squared.
And then finally we have
the function f of x equals
two times the quantity x
minus one squared plus three.
To graph this function,
we take the graph of the green function,
which is f of x equals
two times the quantity
x minus one squared, and
then because k is three,
we shift the graph up three units.
So here we have the green function,
f of x equals two times the
quantity x minus one squared.
We shift this up three units
which gives us the purple graph
which is the given
function f of x equals two
times the quantity x minus
one squared plus three.
One other important
property to notice here
is that the vertex of
a parabola in this form
is always going to be h comma k.
Notice in our function,
h is one and k is three,
and the vertex is one comma
three, this point here.
I hope you found this helpful.
