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Question 184808This question is from textbook Algebra and Trigonometry Structure and Method book 2
: I have been working on this math problem and I can't seem to figure it out. I was wondering if someone could help me? Please and Thank You!! I would deeply appreciate it!!
Find all intercepts of the parabola y-2=-1/2(x-4)^2 This question is from textbook Algebra and Trigonometry Structure and Method book 2

Found 2 solutions by jojo14344, MathTherapy:
Answer by jojo14344(1513) About Me  (Show Source):
You can put this solution on YOUR website!

Given: y-2=-1%2F2%28x-4%29%5E2--->y-2=%28-1%2F2%29%28x%5E2-8x%2B16%29, distribute -1%2F2 and transfer -2 to the right.
y=%28-1%2F2%29x%5E2-%28-8%2F2%29x%2B%28-16%2F2%29%2B2=%28-1%2F2%29x%5E2%2B4x-8%2B2
y=%28-1%2F2%29x%5E2%2B4x-6, EQN 1
First, we'll find the vertex thru Vertex form: y=a%28x-h%29%5E2%2Bk%5E2
wheresystem%28h=x%2Ck=y%29 of the vertex
we complete the square:
Divide the whole eqn by -1%2F2:
y%2F%28-1%2F2%29=%28%28-1%2F2%29x%5E2%2B4x-6%29%2F%28-1%2F2%29--->
-2y=x%5E2-8x%2B12
-2y=%28x%5E2-8x%2Bhighlight%2816%29%29%2B12-highlight%2816%29
-2y=%28x-4%29%5E2-4
cross%28-2%29y%2Fcross%28-2%29=%28%28x-4%29%5E2-4%29%2F-2
y=%28-1%2F2%29%28x-4%29%5E2-%284%2F-2%29
y=%28-1%2F2%29%28x-4%29%5E2%2B2
You see, highlight%28h=x=4%29 & highlight%28k=y=2%29--->VERTEX (4,2)
Next:
Let Fx=0, as per EQN 1
y=%28-1%2F2%29%280%29%5E2%2B4%280%29-6
highlight%28y=-6%29, Y Intercept
Next:
For the X Intercepts, we solve the Eqn1:
by QUADRATIC, wheresystem%28a=-1%2F2%2Cb=4%2Cc=-6%29
Solving for he discriminant: b%5E2-4ac=4%5E2-4%28-1%2F2%29%28-6%29=16-12=4
Therefore,
x=%28-4%2B-sqrt%284%29%29%2F%282%2A-1%2F2%29=%28-4%2B-2%29%2F-1
x=%28-4%2B2%29%2F-1=-2%2F-1=highlight%282%29
Also, x=%28-4-2%29%2F-1=-6%2F-1=highlight%286%29
Then we see the graph, plotting all intercepts:

Thank you,
Jojo

Answer by MathTherapy(10560) About Me  (Show Source):
You can put this solution on YOUR website!
I have been working on this math problem and I can't seem to figure it out. I was wondering if someone could help me? 
Please and Thank You!! I would deeply appreciate it!!

Find all intercepts of the parabola y-2=-1/2(x-4)^2

TOTALLY, TOTALLY UNNECESSARY steps by the person who responded.

y+-+2+=+%28-+1%2F2%29%28x+-+4%29%5E2
y+=+%28-+1%2F2%29%28x+%2B+highlight%28-+4%29%29%5E2+%2B+highlight_green%282%29 ----- Adding 2 to both sides
y+=+a%28x+%2B+highlight%28-+h%29%29%5E2+%2B+highlight_green%28+k%29 ----- Vertex form of a parabola, where the VERTEX, (h, k) = (4, 2)
"h" is the x-coordinate of the vertex of the parabola, is also the paeabola's Axis of Symmetry, as well as the MIDWAY point
between the x-intercepts.
If MiDWAY between 2 points of a HORIZONTAL line is 4, then LEFT and RIGHT endpoints are: 4+-+4%2F2+=+4+-+2+=+2, and 4+%2B+4%2F2+=+4+%2B+2+=+6.
Therefore, x-intercepts are 2 and 6, or at the coordinate points: (2, 0) and (6, 0).


Although more time-consuming, we can also do the following:
y+-+2+=+%28-+1%2F2%29%28x+-+4%29%5E2
y+=+%28-+1%2F2%29%28x+-+4%29%5E2+%2B+2 ---- Adding 2 to both sides
y+=+%28-+1%2F2%29%28x%5E2+-+8x+%2B+16%29+%2B+2
0+=+%28-+1%2F2%29%28x%5E2+-+8x+%2B+16%29+%2B+2 ---- Substituting 0 for y, since y-values on x-axis are 0
- 2(0) = -+2%28%28-+1%2F2%29%28x%5E2+-+8x+%2B+16%29+%2B+2%29%29 ---- Multiplying each side by - 2
matrix%281%2C3%2C+0+=+x%5E2+-+8x+%2B+16+-+4%2C+or%2C+x%5E2+-+8x+%2B+12+=+0%29
(x - 2)(x - 6) = 0 ---- Factorizing the trinomial
x - 2 = 0       OR      x - 6 = 0
x = 2           OR      x = 6
x-INTERCEPTS: (2, 0) (6, 0)