SOLUTION: Find the equation of the parabola with vertex (-2,1) and x-intercepts -4 and 0. Thank you for your help.

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Question 1002197: Find the equation of the parabola with vertex (-2,1) and x-intercepts -4 and 0.
Thank you for your help.
Found 2 solutions by MathLover1, josmiceli:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
given:
vertex (-2,1) and
x-intercepts -4 and 0
the vertex form of equation of the parabola:
y=a%28x-h%29%5E2%2Bk if the vertex is at (-2,1)=(h,k) we have
y=a%28x-%28-2%29%29%5E2%2B1
y=a%28x%2B2%29%5E2%2B1....now use x-intercepts -4 to find a
if x-intercepts -4, we have a point (-4,0)=(x,y)

0=a%28-4%2B2%29%5E2%2B1
0=a%28-2%29%5E2%2B1
0=a%2A4%2B1
4a%2B1=0
4a=-1
a=-1%2F4
so, your equation is:

y=-%281%2F4%29%28x%2B2%29%5E2%2B1
or, in standard form

y=-%281%2F4%29%28x%5E2%2B4x%2B4%29%2B1
y=-%281%2F4%29x%5E2-x-1%2B1
y=-%281%2F4%29x%5E2-x







Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
If the x-intercepts are: +r%5B1%5D+ and +r%5B2%5D+,
Then you can say:
+%28+x+-+r%5B1%5D+%29%2A%28+x+-+r%5B2%5D+%29+=+0+
+x%5E2+-+%28+r%5B1%5D+%2B+r%5B2%5D+%29%2Ax+%2B+r%5B1%5D%2Ar%5B2%5D+=+0+
-------------------
You are given:
( -4,0 ) and ( 0,0 ) , so
+r%5B1%5D+=+-4+
+r%5B2%5D+=+0+
-------------
+x%5E2+-+%28+r%5B1%5D+%2B+r%5B2%5D+%29%2Ax+%2B+r%5B1%5D%2Ar%5B2%5D+=+0+
+x%5E2+-+%28+-4+%2B+0+%29%2Ax+%2B+%28-4%29%2A0+=+0+
+x%5E2+-+4x+=+0+
If the equation has the form:
+a%2Ax%5E2+%2B+b%2Ax+%2B+c+=+0+
+x%5E2+%2B+%28b%2Fa%29%2Ax+=+0+ ( +c+=+0+ )
+b%2Fa+=+4+
------------
The formula for the x-coordinate of the vertex is:
+x%5Bv%5D+=+-b%2F%282a%29+
+x%5Bv%5D+=+-2+
+-b%2F%282a%29+=+-2+
+b%2Fa+=+4+ ( that confirms my answer )
+b+=+4a+
Now I can say:
+y+=+a%2Ax%5E2+%2B+b%2Ax+
solving at ( -2,1 )
+1+=+a%2A%28-2%29%5E2+%2B+b%2A%28-2%29+
+4a+-+2b+=+1+
and since +b+=+4a+
+4a+-+2%2A%28+4a+%29+=+1+
+-4a+=+1+
+a+=+-1%2F4+
and
+b+=+4a+
+b+=+4%2A%28-1%2F4+%29+
+b+=+-1+
---------------------
The equation is:
+y+=+-%28+1%2F4+%29%2Ax%5E2+-+x+
---------------------
check:
(0,0)
+y+=+-%28+1%2F4+%29%2Ax%5E2+-+x+
0+=+-%28+1%2F4+%29%2A0%5E2+-+0+
+0+=+0+
---------------------
(-4,0)
+y+=+-%28+1%2F4+%29%2Ax%5E2+-+x+
+0+=+-%28+1%2F4+%29%2A%28-4%29%5E2+-+%28-4%29+
+0+=+-4+%2B+4+
+0+=+0+
--------------
(-2,1)
+y+=+-%28+1%2F4+%29%2Ax%5E2+-+x+
1+=+-%28+1%2F4+%29%2A%28-2%29%5E2+-%28-2%29+
+1+=+-1+%2B+2+
+1+=+1+
OK