We are asked to write the given
quadratic function in vertex form.
Notice right now the quadratic
function is in general form.
To write the function in vertex form,
we'll use the formula shown
here to determine the vertex.
Once we determine the vertex,
this will give us h and k in vertex form,
and then the value of a in vertex form
is the same as the value
of a in general form.
Let's begin by determining
the values of a, b, and c.
a is the coefficient of
x squared which is three.
b is the coefficient of x.
Because we have minus
10 x, b is negative 10.
c is the constant term.
Because we have minus
five, c is negative five.
The next step is to determine the equation
of the axis of symmetry,
given by x equals negative
b divided by two a.
And because the vertex is
on the axis of symmetry,
this x-value also gives us the
x-coordinate of the vertex.
So for x equals negative b,
divided by two a,
we have b equals negative
10 and a equals three,
which gives us negative negative 10,
or the opposite of negative 10,
divided by two times a
which is two times three.
Simplifying we have positive 10 over six.
Leaving this in fraction form,
we can simplify by
dividing 10 and six by two,
which gives us the equivalent
simplified fraction of 5/3.
Which means now we know the vertex
has an x-coordinate of 5/3,
and therefore h is equal to 5/3.
To determine the
y-coordinate of the vertex,
we need to determine the function value
when x equals 5/3,
which is f of negative b divided by two a.
So for f of 5/3,
we substitute 5/3 for x,
which is equal to three
times the square of 5/3
minus 10 times 5/3
minus five.
Simplifying, the square of 5/3 is 25/9,
giving us three times 25/9.
And then here we can
write 10 as a fraction
with a denominator of one,
giving us minus 50/3.
Let's write minus five
as minus five over one.
Multiplying here let's
write three as a fraction
with a denominator of one.
Multiplying we have 75/9.
We are going to have to
have a common denominator.
Let's use the LCD of nine.
Let's write 50/3 as an equivalent fraction
with a denominator of nine by multiplying
the numerator and denominator by three.
Which gives us minus 150/9.
And then for five over one,
we multiply the numerator
and denominator by nine,
giving us minus 45/9.
Now that we have a common
denominator we can simplify.
The denominator remains nine.
In the numerator we have 75 minus 150
which is negative 75 minus 45
which is negative 120.
We now know f of 5/3
equals negative 120/9,
but this does simplify
because 120 and nine share
a common factor of three.
This simplifies to negative 40/3.
So now we know the
y-coordinate of the vertex
is negative 40/3.
We know k is equal to negative 40/3,
and we already determined a
is equal to positive three.
This is all we need in
order to write the function
in vertex form.
In vertex form we have f of x
equals a which is three
times the quantity x minus h squared,
which in our case gives us
the quantity x minus 5/3 squared
plus k gives us plus negative 40/3.
But plus negative 40/3 is
equivalent to minus 40/3,
so let's rewrite this as f of x
equals three times the
quantity x minus 5/3 squared
minus 40/3.
Let's verify this graphically
by graphing the function
in vertex form.
If we graph the function in vertex form,
this is the graph of the function.
Notice how the vertex is
the point 4/3 comma negative
40/3, this point here.
Because a is positive notice
how the parabola opens up,
and for the last check,
notice how the vertical
intercept or the y-intercept
is negative five,
which in general form is the value of c.
Notice how here we have
c equals negative five.
I hope you found this helpful.
