The quadratic function f(x)= -1/2(x+3)^2+1. The point (1,1) in the graph of y=x^2 is transformed to which point on the graph of y= -1/2(x+3)^2+1?
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A. (-2,1/2)
B. (-2,3/2)
C. (-2,-1)
D. (5/2,2)
to get from y=x^2 to y= -1/2(x+3)^2+1
reflect across the x-axis, scale by 1/2 in the y-direction, shift left 3, shift up 1.
(1,1) -> (1,-1) -> (1,-1/2) -> (-2,-1/2) -> (-2,1/2)
To find the transformation of the point (1, 1) in the graph of y = x^2 to the graph of y = -1/2(x + 3)^2 + 1, we need to substitute x = 1 into the quadratic function.
First, let's determine the value of y in the original equation y = x^2 when x = 1:
y = (1)^2 = 1
Now, let's substitute x = 1 into the transformed equation y = -1/2(x + 3)^2 + 1:
y = -1/2(1 + 3)^2 + 1
y = -1/2(4)^2 + 1
y = -1/2(16) + 1
y = -8 + 1
y = -7
Therefore, the transformed point is (1, 1) --> (-2, -7).
From the answer choices, none of them match the transformed point (-2, -7). So, none of the given answer choices is correct.
To find the point on the graph of the quadratic function y = -1/2(x + 3)^2 + 1 that corresponds to the point (1, 1) on the graph of y = x^2, we need to substitute the x-coordinate of the given point into the function and solve for the y-coordinate.
Given:
Quadratic function: f(x) = -1/2(x + 3)^2 + 1
Point on graph y = x^2: (1, 1)
Substitute x = 1 into the function f(x):
f(1) = -1/2(1 + 3)^2 + 1
= -1/2(4)^2 + 1
= -1/2(16) + 1
= -8 + 1
= -7
Therefore, the point on the graph of y = -1/2(x + 3)^2 + 1 that corresponds to the point (1, 1) is (1, -7).
Now let's find the transformed point on the graph of y = -1/2(x + 3)^2 + 1.
To transform the point (1, -7) on the graph of y = -1/2(x + 3)^2 + 1, we need to shift the point horizontally by 3 units to the left.
Given:
Original point: (1, -7)
Horizontal shift: 3 units to the left
To shift the point, subtract 3 from the x-coordinate:
x' = x - 3
= 1 - 3
= -2
Therefore, the transformed point on the graph of y = -1/2(x + 3)^2 + 1 corresponding to the point (1, 1) on the graph of y = x^2 is (-2, -7).
Hence, the correct option is C. (-2, -1).