RA to find the vertex focus in direct Xs
for the parabola given by the following equation,
x squared plus eight x minus 20 Y must 24 equals zero.
Define the vertex focus in direct.
Xs will write the equation of the parabola in standard form,
which is one of the two forms shown below.
Because our equation contains x squared
and not y squared, we'll write the equation in the first
form or the square
of x minus H equals four P times in quantity Y minus K.
Because the X part is squared, the probla will open up
or down based upon the sign of P.
If P is positive, it opens up.
If P is negative, it opens down
and H comm K is the vertex to write the given equation.
In this form, we first isolate the X terms on the left
and move the other terms to the right.
This indicates the first step is to add 20 Y
and add 24 to both sides of the equation.
And now I complete the square on the left side
to make the left side a perfect square.
Trinomial recall, to do this, we add the square of half of B
or half of the coefficient of x, which is eight
to both sides of the equation.
This indicates we add the square of four
or 16 to both sides of the equation,
and now we simplify and factor.
On the left we have a perfect square trinomial.
The factors of 16 that add to eight or four
and four giving us two factors of x plus four.
On the right simplifying, we have 20 y plus 40,
and now we can factor 20 from the right,
which gives us the square
of x plus four equals 20 times the quantity y plus two.
The equation is now in standard form for apla.
Again, the X part is squared indicating
that parabola will open up or down, but
because four P is equal to 20 P is positive, we now know
that pela will open up
and will resemble the parabola shown here.
On the far right in our notes recall,
the vertex is H comma K,
and the absent IFP will give us a distance from the vertex
to the focus as well as a distance from the vertex to
the DirectX, this horizontal line here.
So for the next step, let's go ahead
and find the vertex, which is H comma K.
Because we have x plus four,
the X coordinate is negative four
because we have Y plus two.
The Y coordinate is negative two.
The vertex is the point negative four comma negative two.
Let's go ahead and plot this on the coordinate plane.
Now remember, we know the pbla opens up
and we know the focus.
This orange point here is always inside the parabola
and the distance from the vertex to the focus
is always equal to the opposite bio P.
In our case, P is positive though, and
therefore if we find P,
we can find the distance from the vertex to the focus.
Going back to our equation, notice four P is equal to 20,
so four P is equal to 20.
We divide both sides by four and we have P equals five.
We now know the focus is five units above the vertex,
which means to find the focus, we add five
to the Y coordinate of the vertex,
or just moving up five units from the vertex.
We can see the focus is going to be this point here,
negative four comma three plus's.
Go ahead and find this more formally.
The focus is the point negative four
comma negative two plus five.
Simplifying we have the focus is the point
negative four comma three.
Now going back to the graph,
we know the direct XX is a horizontal line that is P units
below the vertex or five units below the vertex.
Let's go ahead and sketch that.
The horizontal line of five units
below the vertex is y equals negative seven.
This horizontal line here
More formally defined the equation of the DirectX.
We would subtract five from the Y coordinate of the vertex
giving us Y equals negative two minus five
or Y equals negative seven.
And before we go, let's go ahead and graph the apla.
The APLA would look something like this.
I hope you found this helpful.