Textbook Question
Begin by graphing the cube root function, f(x) = ∛x. Then use transformations of this graph to graph the given function. g(x) = (1/2)∛(x+2) - 2
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3. Functions
Transformations
2:25 minutesProblem 15
Textbook Question
Use the graph of y = f(x) to graph each function g.
g(x) = −ƒ( x/2) +1
Verified step by step guidance1
Step 1: Analyze the given graph of y = f(x). The graph is a horizontal line segment with endpoints at (1, -3) and (4, -3). This indicates that f(x) = -3 for all x in the interval [1, 4].
Step 2: Understand the transformation g(x) = -f(x/2) + 1. This involves three transformations: (1) horizontal scaling by a factor of 2 (x/2), (2) reflection across the x-axis (-f), and (3) vertical shift upward by 1 (+1).
Step 3: Apply the horizontal scaling. Replace x with x/2 in f(x). This stretches the graph horizontally by a factor of 2. The new domain of f(x/2) becomes [2, 8] because the original domain [1, 4] is scaled by 2.
Step 4: Apply the reflection across the x-axis. Multiply the function values by -1. Since f(x/2) = -3 for x in [2, 8], the reflected values become +3 for the same interval.
Step 5: Apply the vertical shift. Add 1 to the reflected values. The final function g(x) = -f(x/2) + 1 will have values of 3 + 1 = 4 for x in [2, 8]. The graph of g(x) is a horizontal line segment at y = 4 over the interval [2, 8].
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Transformation
Function transformations involve altering the graph of a function through shifts, stretches, compressions, or reflections. In this case, the function g(x) = -f(x/2) + 1 reflects the original function f(x) across the x-axis, stretches it horizontally by a factor of 2, and shifts it upward by 1 unit.
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Domain & Range of Transformed Functions
Horizontal Stretch
A horizontal stretch occurs when the input of a function is multiplied by a factor less than 1. For g(x) = -f(x/2), the x-values of f(x) are effectively doubled, which means that the graph of g will spread out horizontally, making it wider compared to f.
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Reflection Across the X-axis
Reflection across the x-axis changes the sign of the output values of a function. In g(x) = -f(x/2), this reflection means that if f(x) has positive values, g(x) will have corresponding negative values, effectively flipping the graph of f over the x-axis.
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