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What is the vertex and inverse of this parabola? What am I doing wrong? HELP!
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# What is the vertex and inverse of this parabola? What am I doing wrong? HELP!

[From: ] [author: ] [Date: 11-05-02] [Hit: ]
given that x This is what I have so far....4 = -x^2 + 4x +44 = (-x+2)^2 (completing the square0 = -1(x-2)^2 - 4 (putting into standard form)Vertex: (2, -4)Opens downy intercept: -8 (solving for x = 0)Domain: xNow,......
321. I need to find the vertex of this parabola, as well as the domain and range. I also need to find its inverse....

y = 4x - x^2, given that x
This is what I have so far....

4 = -x^2 + 4x +4
4 = (-x+2)^2 (completing the square
0 = -1(x-2)^2 - 4 (putting into standard form)

Vertex: (2, -4)
Opens down
y intercept: -8 (solving for x = 0)
Domain: x
Now, for the inverse.... this is what I have....

x = -1(y-2)^2 - 4
-x = (y-2)^2 + 4 (multiplying by -1 on both sides)
-x - 4 = (y-2)^2 (moving 4)
sqrt(-x - 4) = y - 2 (rooting both sides)

OK, but here I am stuck because I have a square root of a negative number and I know that this problem does not deal with that stuff.... well, at least I don't think (or hope?) so.... Can you help me?

I need to find the inverse function, and the domain/range of the inverse? Also, I need to graph them both on the same plane, but you cant show me that on Y!A,.... so if you can help, please help me. Thanks!

-
it's hard to properly complete the square
unless coefficient of x = 1
so
y = -x² + 4x
-y = x² - 4x
-y + 4 = x² - 4x + (-2)²
-y + 4 = (x - 2)²
-y = (x - 2)² - 4
y = -(x - 2)² + 4 ← see, you got the signs wrong
so the vertex is actually (2, 4)
does open down
or, in other words, vertex is max
so
range is y ≤ 4

for the inverse,
I prefer to swap vars last, so
y = -(x - 2)² + 4
4 - y = (x - 2)²
±√(4 - y) = x - 2
2 ± √(4 - y) = x
now, since you started with only the left side of the parabola,
you should end up with only the bottom half in its inverse:
2 - √(4 - y) = x
so
y = 2 - √(4 - x)

for the inverse domain and range,
just swap the domain and range of the original function:
D: x ≤ 4
R: y ≤ 2

see
http://s584.photobucket.com/albums/ss282…
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