If you horizontally stretch the square root parent function, F(x) = √√x, by ³/ units, what is the equation of the new function?
A. G(X) = √√x+3
B. G(x) =
C. G(x) = ³√x
D. G(x)=√x+44
The Correct Answer and Explanation is:
The problem asks about the transformation of the square root function ( f(x) = \sqrt{x} ) through a horizontal stretch by a factor of ( \frac{3}{2} ).
Understanding the Transformation
To address the question, we need to understand how a horizontal stretch affects a function. A horizontal stretch by a factor of ( c ) is achieved by replacing ( x ) with ( \frac{x}{c} ) in the function’s equation. This transformation causes the graph to stretch away from the y-axis, making it “wider.”
For example, if you stretch the basic square root function ( f(x) = \sqrt{x} ) horizontally by a factor of ( \frac{3}{2} ), you replace ( x ) with ( \frac{x}{\frac{3}{2}} ), or equivalently ( \frac{2}{3}x ).
Transformation of ( f(x) = \sqrt{x} )
The original function is ( f(x) = \sqrt{x} ). A horizontal stretch by a factor of ( \frac{3}{2} ) means we replace ( x ) with ( \frac{2}{3}x ). The new function will be:
[
g(x) = \sqrt{\frac{2}{3}x}
]
This transformation makes the graph of the function ( f(x) ) wider, stretching it along the x-axis by a factor of ( \frac{3}{2} ).
Analyzing the Choices
Now, let’s look at the given options and analyze which one matches the transformed function:
- A. ( G(x) = \sqrt{x+3} ): This is a horizontal shift (right by 3 units), not a stretch.
- B. ( G(x) = ): This is incomplete and cannot be correct.
- C. ( G(x) = \sqrt[3]{x} ): This represents a cubic root function, not a square root function, so it is not the correct transformation.
- D. ( G(x) = \sqrt{x} + 44 ): This represents a vertical shift (up by 44 units), not a horizontal stretch.
Conclusion
None of the provided answers exactly match the horizontal stretch by a factor of ( \frac{3}{2} ). However, based on the given options, none would correctly reflect the transformation as described (i.e., the answer should be ( G(x) = \sqrt{\frac{2}{3}x} )). The correct transformation, if given as an option, would look something like:
[
G(x) = \sqrt{\frac{2}{3}x}
]
Thus, based on the given options, we must conclude that none of them represent the correct transformation, but the equation should have been ( G(x) = \sqrt{\frac{2}{3}x} ).