SOLUTION: Vertex (1,2) Point (3,-4) Write the equation of the graph below in standard form y=a(x-h)^2+k and general form y=ax^2+bx+c. I managed to solve for the standard form (which I w

Algebra ->  Graphs -> SOLUTION: Vertex (1,2) Point (3,-4) Write the equation of the graph below in standard form y=a(x-h)^2+k and general form y=ax^2+bx+c. I managed to solve for the standard form (which I w      Log On


   

Question 1200626: Vertex (1,2) Point (3,-4)
Write the equation of the graph below in standard form y=a(x-h)^2+k and general form y=ax^2+bx+c.
I managed to solve for the standard form (which I will show) but I am having difficulties transferring it to general form.
y=a%28x-h%29%5E2%2Bk
-4=a%283-1%29%5E2%2B2
-4=4a+2
-2 -2
-6=4a
-- --
4 4
y=-%286%2F4%29%28x-1%29%5E2%2B2
Does that look correct? If so how I mold it into general form?
Found 2 solutions by josgarithmetic, math_tutor2020:
Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
Vertex (1,2) Point (3,-4)
y=a(x-h)^2+k
------
------

y=a%28x-1%29%5E2%2B2
and_then
-4=a%283-1%29%5E2%2B2
-4=a%2A%282%29%5E2%2B2
-4=4a%2B2
4a=-6
a=-3%2F2
-
highlight%28y=-%283%2F2%29%28x-1%29%5E2%2B2%29

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Looks good so far.
I appreciate you showing your work.

The -6/4 can be reduced to -3/2.

This means the vertex form is
y+=+%28-3%2F2%29%28x-1%29%5E2%2B2
where
a = -3/2
h = 1
k = 2

Let's expand things out (using the FOIL rule) and combine like terms
y+=+%28-3%2F2%29%28x-1%29%5E2%2B2

y+=+%28-3%2F2%29%28x-1%29%28x-1%29%2B2

y+=+%28-3%2F2%29%28x%5E2-1x-1x%2B1%29%2B2 FOIL rule

y+=+%28-3%2F2%29%28x%5E2-2x%2B1%29%2B2

y+=+%28-3%2F2%29%28x%5E2%29%2B%28-3%2F2%29%28-2x%29%2B%28-3%2F2%29%281%29%2B2 Distribute

y+=+%28-3%2F2%29x%5E2%2B3x-3%2F2%2B2

y+=+%28-3%2F2%29x%5E2%2B3x-3%2F2%2B4%2F2

y+=+%28-3%2F2%29x%5E2%2B3x%2B%28-3%2B4%29%2F2

y+=+%28-3%2F2%29x%5E2%2B3x%2B1%2F2
The equation is now in general form y = ax^2+bx+c, where,
a = -3/2
b = 3
c = 1/2

--------------------------------------
Check:
Plugging x = 1 should lead to y = 2
y+=+%28-3%2F2%29x%5E2%2B3x%2B1%2F2
y+=+%28-3%2F2%29%281%29%5E2%2B3%281%29%2B1%2F2
y+=+%28-3%2F2%29%281%29%2B3%281%29%2B1%2F2
y+=+-3%2F2%2B3%2B1%2F2
y+=+-3%2F2%2B6%2F2%2B1%2F2
y+=+%28-3%2B6%2B1%29%2F2
y+=+4%2F2
y+=+2
Therefore, (1,2) is on the parabola.

Plugging x = 3 should lead to y = -4
y+=+%28-3%2F2%29x%5E2%2B3x%2B1%2F2
y+=+%28-3%2F2%29%283%29%5E2%2B3%283%29%2B1%2F2
y+=+%28-3%2F2%29%289%29%2B3%283%29%2B1%2F2
y+=+-27%2F2%2B9%2B1%2F2
y+=+-27%2F2%2B18%2F2%2B1%2F2
y+=+%28-27%2B18%2B1%29%2F2
y+=+-8%2F2
y+=+-4
That confirms (3,-4) is also on the parabola.
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Graph:

Desmos and GeoGebra are two graphing options I recommend.

Here is the link to the interactive Desmos graph.
https://www.desmos.com/calculator/vz0x6inti1
Next to each equation (on the left side panel) are red and blue buttons. Click each of those to turn off/on the equation curve.
This will help illustrate the two curves occupy the same exact space. They pass through the same set of points.
This is visual confirmation that y+=+%28-3%2F2%29%28x-1%29%5E2%2B2 is the same as y+=+%28-3%2F2%29x%5E2%2B3x%2B1%2F2

Side note: The value of a = -3/2 is negative, which means the parabola opens downward.