We are asked to find the equation
of the quadratic function in the form
f of x equals a times the quantity x minus h squared plus k
based upon the graph.
We will also find the equation in the form
f of x equals a x squared plus bx plus c as well.
For this form of a quadratic function,
h comma k is the vertex.
So looking at the graph, notice how the vertex,
because the parabola opens down, is the highest point
on the graph, this point here,
where the ordered pair is negative two comma four.
So because the vertex is negative two comma four,
we know that h is equal to negative two
and k is equal to positive four.
Performing the substitution gives us f of x
equals a times the quantity x minus negative two squared
plus k, which gives us plus four.
And x minus negative two simplifies to x plus two.
So we have f of x equals a times the quantity
x plus two squared plus four.
And now we need to find another point on the graph
and perform substitution to determine the value of a.
Notice how this parabola
has two convenient horizontal intercepts.
It passes through the origin or the ordered pair
zero comma zero as well as this point here
with the ordered pair negative four comma zero.
Let's use this ordered pair and perform substitution
to find the value of a.
If the point negative four comma zero is on the graph
of the function, then we know f of negative four
must equal zero.
So we can substitute negative four for x
and zero for the function value and then solve for a.
Performing the substitution gives us zero equals a
times the quantity negative four plus two squared plus four.
And now simplifying, negative four plus two
is negative two.
The square of negative two is positive four
and positive four times a is four a,
giving us zero equals four a plus four.
And now to solve for a, we subtract four on both sides,
which gives us negative four equals four a
and then we divide both sides by four.
Simplifying, we have a is equal to negative one
and therefore the function must be f of x
equals a, which is negative one, times the quantity
x plus two squared plus four.
Sometimes instead of a negative one though,
we'll just see a negative.
We can also write f of x as f of x
equals negative or the opposite of the quantity
x plus two squared plus four.
Now that we have the function in this form,
let's also write the function in the form
f of x equals a x squared plus bx plus c.
To do this, we'll have to square the binomial,
take the opposite, and then add four.
And there's no shortcuts to squaring the binomial.
We have two factors of x plus two.
When multiplying two binomials, we have four products.
We distribute the x and we distribute the two.
So we have f of x equals the opposite of
the quantity x times x is x squared,
then x times two gives us two x,
but then the next product is two times x
which is also two x, two x plus two x gives us plus four x,
and then two times two is four,
giving us plus four, and then we still have plus four
outside the parentheses.
And now to clear the parentheses here,
we can distribute the negative or if it's helpful,
think of distributing a negative one,
which gives us f of x
equals negative x squared minus four x
minus four plus four.
And negative four plus four is equal to zero,
giving us f of x
equals negative x squared minus four x.
So now we have the equation of the function
in both forms of a quadratic function.
This first form is often called vertex form
and this last form is sometimes called standard form
or general form.
I hope you found this helpful.