Question 1179671: For the equation, identify the equation of the axis of symmetry, the coordinates of the vertex, the y-intercept, the zeros, and the maximum or minimum values.
1. y = -2(x - 3) 2 + 9
Found 4 solutions by Boreal, josgarithmetic, greenestamps, ewatrrr:
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! The vertex is at (-h, k), which would be x=3, y=9 or (3, 9)
The axis of symmetry is therefore at x=3.
the y-intercept is where x=0, and that is -2(9)+9=-9 so (0, -9)
The maximum value for this convex upward parabola is when x=3 and that has already been shown to have y=9.
The minimum values are at -oo.
For the zeros, let y=0,
then -2(x-3)^2=-9
or (x-3)^2=9/2
x-3=+/- 3(sqrt(2)/2
so the zeros are 3+/- (3/2) sqrt (2)
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Can check by writing it in standard form
y=-2x^2+12x-18+9; -2x^2+12x-9=0, or 2x^2-12x+9=0
x= (1/4)(12+/-sqrt(144-72)) or (1/4)(12+/-6 sqrt(2))
roots are 3 +/- (3/2) sqrt(2)
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Answer by josgarithmetic(39617)  (Show Source):
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
Note, for future reference: In typed text, use "^" to denote an exponent. In your problem, "y=-2(x-3)^2+9" instead of "y=-2(x-3)2+9".
(1) Vertex and axis of symmetry: The given equation is in vertex form: y=a(x-h)^2+k
In that form, the vertex is at (h,k). And the axis of symmetry is the vertical line through the vertex.
ANSWER: The vertex is (3,9); the axis of symmetry is x=3.
(2) Maximum and minimum: The coefficient on the x^2 term is negative, so the graph is a parabola that opens downward. So it has a maximum but no minimum. The maximum is at the vertex.
It's easy to calculate the maximum value when the equation is in vertex form -- it's at the vertex (i.e., when the (x-3)^2 part of the equation is equal to 0).
ANSWER: maximum value at (3,9); no minimum value.
(3) y-intercept: When x=0. -2(-3)^2+9 = -2(9)+9 = -18+9 = -9.
ANSWER: y-intercept (0,-9)
(4) zeros (x-intercepts): When y=0. Put the equation in standard form. If it can be factored, then finding the zeros is simple; if not, use the quadratic formula.
-2(x-3)^2+9 = -2(x^2-6x+9)+9 = -2x^2+12x-9
That does not factor, so use the quadratic formula.
Answer by ewatrrr(24785)  (Show Source):
You can put this solution on YOUR website!
Hi
the vertex form of a Parabola opening up(a>0) or down(a<0),
where(h,k) is the vertex and x = h is the Line of Symmetry
y = -2(x - 3)^2 + 9
identify:
axis of symmetry, x = 3
the coordinates of the vertex, V(3,9)
the y-intercept, P(0,-9)
the zeros, x= 3 ± 3/√2 0r x = 3 ± 3√2/2
and the maximum or minimum values. y = 9
Wish You the Best in your Studies.
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