In this problem, we're 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
where when we have a quadratic function in this form,
h comma k is the vertex.
We will also find the equation in the form
f of x equals a x squared plus bx plus c for practice.
So looking at the graph, notice how the vertex
is the lowest point on the graph
because the parabola opens up.
This point is the vertex where the ordered pair
is four comma negative three.
So because the vertex is four comma negative three,
we know that h is four and k is negative three.
Let's perform the substitution into the quadratic function.
This gives us f of x
equals a times the quantity x minus four squared
and then plus negative three or just minus three.
And now to find the value of a,
we need to select another point on the graph
and then perform substitution so that we can solve for a.
So we need to find a convenient point on the graph
where the coordinates are integers.
For example, this point here where the ordered pair
is two comma five.
So because the point two comma five is a point on the graph,
we know that f of two is equal to five.
So if f of two is equal to five,
we can substitute two for x, five for the function value,
and then solve for a.
So performing the substitution, we would have five
equals a times the quantity
two minus four squared minus three.
And now simplifying, two minus four is negative two.
The square of negative two is positive four.
Four times a is four a, so we have five equals
four a minus three.
To solve for a, we add three to both sides.
Five plus three is eight, giving us eight equals four a.
We divide both sides by four.
Simplifying, we have a equals two.
Now that we know a is equal to two,
we know the function must be f of x
equals two times the quantity x minus four
squared minus three.
So this is the function in vertex form
or the form f of x equals a times the quantity
x minus h squared plus k.
But let's also find the function in this form.
To do this, we'll have to square x minus four,
multiply it by two, and then subtract three.
And there's no shortcut for squaring x minus four.
Let's write out two factors of x minus four.
We have f of x equals two times the quantity
x minus four times the quantity x minus four minus three.
And now to multiply the two binomials,
we will have four products.
We distribute the x and we distribute the negative four.
Which gives us f of x equals two times the quantity
x times x is squared, and then we have x times
negative four which is negative four x,
but then the next product is negative four times x,
which is also negative four x.
Negative four x minus four x is negative eight x,
giving us minus eight x.
And then negative four times negative four is 16,
giving us plus 16
and then we still have minus three outside the parentheses.
And now we distribute the two
and then combine like terms.
This gives us f of x equals two x squared
minus 16x plus 32 minus three,
which gives us f of x equals two x squared
minus 16x and 32 minus three is 29,
giving us plus 29.
So depending on your textbook,
this is sometimes called vertex form or standard form
and this form can also sometimes be called standard form
or general form.
So again, it does depend upon your text.
I hope you found this helpful.