Textbook Question
Use the graph of y = f(x) to graph each function g. g(x)=2f(x-1)
3. Functions
Common Functions
1:46 minutesProblem 5
Textbook Question
Use the graph of y = f(x) to graph each function g.
g(x) = f(x-1) - 2
Verified step by step guidance1
Identify the original function f(x) from the graph. Here, f(x) is a horizontal line segment from x = 1 to x = 4 with a constant value of y = -3.
Understand the transformation in g(x) = f(x - 1) - 2. The term (x - 1) inside the function shifts the graph of f(x) horizontally to the right by 1 unit.
The '- 2' outside the function shifts the graph vertically downward by 2 units.
Apply the horizontal shift: move the original segment from [1, 4] to [2, 5] because each x-value increases by 1.
Apply the vertical shift: subtract 2 from the y-value of -3, resulting in a new y-value of -5. So, the new segment for g(x) is from (2, -5) to (5, -5).
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Transformation
Function transformations involve shifting, stretching, or reflecting the graph of a function. In this problem, g(x) = f(x-1) - 2 represents a horizontal shift to the right by 1 unit and a vertical shift downward by 2 units of the original function f(x). Understanding these shifts helps in accurately graphing the new function.
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Domain & Range of Transformed Functions
Horizontal Shift
A horizontal shift occurs when the input variable x is replaced by (x - h), shifting the graph h units to the right if h is positive, or to the left if h is negative. Here, f(x-1) shifts the graph of f(x) one unit to the right, moving every point on the graph accordingly.
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Vertical Shift
A vertical shift involves adding or subtracting a constant to the function's output, moving the graph up or down. In g(x) = f(x-1) - 2, subtracting 2 shifts the entire graph of f(x-1) down by 2 units, lowering the y-values of all points by 2.
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