Which situation best describes the transformation between f(x) = 10x and g(x)=-2 * 10x?
A. The graph of g(x)=-2 * 10x is transcribed 2 units to the right.
B. The graph of g(x)=-2 * 10x is transcribed 2 units to the left.
C. The graph of g(x)=-2 * 10x is compressed vertically and reflected across the x-axis.
D. The graph of g(x)=-2 * 10x is stretched vertically and reflected across the x-axis.
Answer (2)
The correct transformation from f ( x ) = 1 0 x to g ( x ) = − 2 ⋅ 1 0 x involves stretching the graph vertically by a factor of 2 and reflecting it across the x-axis. Thus, the correct answer is option C: The graph of g ( x ) is stretched vertically and reflected across the x-axis.
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The function g ( x ) = − 2 ⋅ 10 x is obtained from f ( x ) = 10 x by multiplying by − 2 .
The absolute value of the factor, ∣ − 2∣ = 2 , stretches the graph vertically by a factor of 2.
The negative sign reflects the graph across the x-axis.
The graph of g ( x ) is stretched vertically and reflected across the x-axis. The answer is The graph of g ( x ) = − 2 ⋅ 10 x is stretched vertically and reflected across the x-axis.
Explanation
Understanding the Problem We are given two functions, f ( x ) = 10 x and g ( x ) = − 2 ⋅ 10 x . We need to determine how the graph of f ( x ) is transformed to obtain the graph of g ( x ) .
Analyzing the Transformation The function g ( x ) is obtained from f ( x ) by multiplying by − 2 . This factor affects the graph of f ( x ) in two ways:
Vertical Stretch/Compression: The absolute value of the factor, ∣ − 2∣ = 2 , determines whether the graph is stretched or compressed vertically. Since 1"> 2 > 1 , the graph is stretched vertically by a factor of 2.
Reflection across the x-axis: The negative sign indicates that the graph is reflected across the x-axis.
Describing the Transformation Therefore, the graph of g ( x ) = − 2 ⋅ 10 x is obtained from the graph of f ( x ) = 10 x by stretching it vertically by a factor of 2 and reflecting it across the x-axis.
Conclusion The correct answer is: The graph of g ( x ) = − 2 ⋅ 10 x is stretched vertically and reflected across the x-axis.
Examples
Imagine you are adjusting the volume and flipping the audio of a sound wave. Multiplying a function by a number greater than 1 stretches it vertically, like increasing the volume of a sound wave. Multiplying by a negative number reflects it across the x-axis, similar to flipping the audio signal. Understanding these transformations helps in various applications, such as image processing, signal analysis, and physics, where functions are manipulated to achieve desired outcomes.
