let g(x)=root25-x^2. The range of a function that will translate the graph of 6 units to the right and 2 units up is ____<=y<=______
Fill in the blank
range of g(x) is [0,5]
so, range of new g'(x) is [2,7]
the horizontal shift does not matter.
To find the range of the translated function, we first need to determine the range of the original function g(x) = √(25-x^2).
The original function is a semicircle with a radius of 5 centered at the origin. The range is the set of all possible y-values that the function can take.
Since the graph is a semicircle, the lowest point on the graph is at the bottom of the semicircle, which is -5. The highest point on the function is at the top of the semicircle, which is 5.
Therefore, the range of the original function is -5 <= y <= 5.
Now, if we translate the graph of the function 6 units to the right and 2 units up, the new graph will have the same range, but shifted vertically by 2 units and horizontally by 6 units.
So the range of the translated function is -5 + 2 <= y <= 5 + 2, which simplifies to -3 <= y <= 7.
Therefore, the range of the translated function is -3 <= y <= 7.
To find the range of the translated function, we need to determine the minimum and maximum values of y after the translation.
First, let's analyze the original function g(x) = √(25 - x^2). The graph of this function is a half-circle centered at (0,0) with a radius of 5.
The range of g(x) is y ≤ 5, since the maximum value of y occurs when x = 0 (at the top of the half-circle).
Now, let's consider the translated function. Translating the function 6 units to the right will shift the graph horizontally to the right by 6 units. Meanwhile, translating it 2 units up will shift the graph vertically up by 2 units.
As a result, the new function will have the same shape as the original, but it will be displaced 6 units to the right and 2 units up.
Therefore, the new range will be y ≤ 5 + 2 = 7.
Hence, the range of the translated function is y ≤ 7.