00:01
In this problem, we have a parabola, and we're given its two x intercepts and its vertex.
00:06
Our job is to find the three forms of the given parabola, that is the vertex form, the standard form, and the factored form.
00:15
So we have our x intercepts, and we have our vertex, and notice that we have the x coordinate of the vertex as the midpoint, or the average, if you will, the mean of the two x coordinates of the x intercept.
00:27
So we have that as 29 .875 instead of rounding.
00:32
Use these three points to find those three forms of the parabola.
00:37
Since we have the vertex, we will start with the vertex form.
00:45
Now we know in general that the vertex form of a parabola can be written with the equation y equals a times x minus h squared plus k, with the vertex is given by h comma k.
01:05
So if you know h and k, we just have to solve for this constant a.
01:12
So here, the equation of the vertex form will be, y equals a times x minus 29 .875, quantity squared, plus 41 .5.
01:28
And how to solve for a, we can use either of the x intercepts.
01:34
Each one will produce the same value for a.
01:37
We'll just pick 0 -0, since that's easy to substitute.
01:42
So when y is 0 and x is 0, that allows us to solve the resulting equation for a.
01:50
So let's make those substitutions here.
01:58
So we see that negative 41 .5 equals a times 29 .875 squared.
02:11
And therefore the exact value of a is negative 41 .5 over 29 .875 squared.
02:23
We can also round that.
02:24
That's approximately negative 0 .0465.
02:32
So now that we have a, we have our vertex form of the given parabola.
02:40
So that vertex form is approximately, y is approximately our a, which is approximately negative 0 .0465 times x minus 29 .875 quantity squared.
03:00
Plus 41 .5.
03:03
So here is our vertex form, the first of our three forms that we're looking for.
03:13
So now that we have the vertex form, we can find the standard form by simply multiplying this out.
03:21
So to get the standard form, let's take our y here.
03:30
We'll use the original a without rounding.
03:34
So here's the a, negative 41 .5, over 29 .875 squared.
03:42
And we want to expand this part.
03:45
So we will get x squared.
03:49
Minus, we get minus 29 .875x twice...