The function 𝑓(𝑥) is transformed to 𝑔(𝑥) by a horizontal shift by 𝑎 units to the left and vertical stretch by 𝑏 units.Under this transformation, a point 𝐴(2,1) on the graph of 𝑓(𝑥) is transformed to a point 𝐵(0,3) on the graph of 𝑔(𝑥).Find the transformed function 𝑔(𝑥).
Question
The function 𝑓(𝑥) is transformed to 𝑔(𝑥) by a horizontal shift by 𝑎 units to the left and vertical stretch by 𝑏 units.Under this transformation, a point 𝐴(2,1) on the graph of 𝑓(𝑥) is transformed to a point 𝐵(0,3) on the graph of 𝑔(𝑥).Find the transformed function 𝑔(𝑥).
Solution
To find the transformed function g(x), we first need to understand how the transformations work.
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A horizontal shift of 'a' units to the left means that every x-coordinate in the original function f(x) will be replaced with (x+a). In this case, since point A(2,1) is transformed to B(0,3), we can see that the x-coordinate has decreased by 2 units. Therefore, 'a' is 2.
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A vertical stretch by 'b' units means that every y-coordinate in the original function f(x) will be multiplied by 'b'. In this case, since the y-coordinate has increased from 1 to 3, 'b' is 3.
So, if the original function is f(x), the transformed function g(x) will be g(x) = 3f(x+2).
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