Example Question - transformation of functions
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Understanding Transformation of Functions
The function f(x) = x^2 is given, and the function g(x) is defined as g(x) = 4f(x) - 4 = 4(x^2) - 4. The transformation from f(x) to g(x) involves two steps: 1. The term 4f(x) indicates that the function f(x) is multiplied by 4, which stretches or scales the graph vertically by a factor of 4. This means that for any x, the value of g(x) will be four times the value of f(x), but the shape of the graph remains the same (it will just be stretched). 2. The term -4 at the end of the expression indicates that the graph is then shifted down by 4 units. It is a vertical translation downwards. Putting these together, we get: A vertical stretch of the graph of f(x) by a factor of 4, followed by a downward shift of the graph by 4 units. The correct answer to the question in the image is D: The graph of g(x) is the graph of f(x) shifted down 4 units. The option about stretching is incorrect because the vertical multiplication by 4 does not stretch the graph of f(x) by a factor of 1/4; it stretches it by a factor of 4. The combination of these transformations makes D the best choice among the provided options.
Transformation of Functions
The equation provided suggests a transformation of the base function \( f(x) \) applied to produce the new function \( y \). The transformation can be described as follows: 1. Horizontal shift: The expression \( (x - 5) \) indicates that the graph of \( f(x) \) is shifted to the right by 5 units. Therefore, \( B = 5 \) (right). 2. Vertical shift: The \( +8 \) at the end of the function indicates that the graph is shifted upwards by 8 units. So, \( D = 8 \) (up). There is no indication in the equation of a horizontal stretch/shrink or reflection (which would be indicated by a multiplier in front of the \( x \) term inside the function), nor is there a vertical stretch/shrink or reflection (which would be indicated by a coefficient in front of the \( f(x) \) term). Thus, we can assume no changes have been made in these aspects. Consequently, \( A \) (representing horizontal stretch/shrink and reflection) and \( C \) (representing vertical stretch/shrink and reflection) remain unchanged: 3. \( A = 1 \) (no horizontal stretch/shrink or reflection). 4. \( C = 1 \) (no vertical stretch/shrink or reflection). Transformations: - A horizontal shift to the right by 5 units. - A vertical shift upwards by 8 units. - No horizontal stretch/shrink or reflection. - No vertical stretch/shrink or reflection. Therefore, your answers will be: A = 1, B = 5, C = 1, D = 8
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