Which of the following describes the graph of [tex]y=\sqrt{-4 x-36}[/tex] compared to the parent square root function?
A. stretched by a factor of 2, reflected over the [tex]x[/tex]-axis, and translated 9 units right
B. stretched by a factor of 2, reflected over the [tex]x[/tex]-axis, and translated 9 units left
C. stretched by a factor of 2, reflected over the [tex]y[/tex]-axis, and translated 9 units right
D. stretched by a factor of 2, reflected over the [tex]y[/tex]-axis, and translated 9 units left
Answer (2)
Rewrite the function: y = − 4 x − 36 = 2 − ( x + 9 ) .
Identify the transformations: reflection over the y-axis, translation 9 units to the left, and vertical stretch by a factor of 2.
Describe the graph: stretched by a factor of 2, reflected over the y-axis, and translated 9 units to the left.
The correct answer is: stretched by a factor of 2, reflected over the y-axis, and translated 9 units left
Explanation
Understanding the Problem We are given the function y = − 4 x − 36 and we want to describe how its graph compares to the parent square root function y = x . This involves identifying any stretches, reflections, and translations.
Rewriting the Function First, we rewrite the given function to make the transformations more apparent. We can factor out − 4 from the expression inside the square root: y = − 4 ( x + 9 ) y = − 4 ⋅ x + 9 y = 4 ⋅ − 1 ⋅ x + 9 y = 2 − ( x + 9 ) Now we can clearly see the transformations.
Identifying the Transformations The transformations are as follows:
Horizontal Reflection: The negative sign inside the square root, i.e., x → − x , indicates a reflection over the y-axis.
Horizontal Translation: The term ( x + 9 ) represents a horizontal translation. Since it's ( x + 9 ) , the translation is 9 units to the left.
Vertical Stretch: The factor of 2 outside the square root represents a vertical stretch by a factor of 2.
Conclusion Combining these transformations, we can describe the graph of y = − 4 x − 36 compared to the parent function y = x as follows: It is stretched by a factor of 2, reflected over the y-axis, and translated 9 units to the left.
Examples
Understanding transformations of functions is useful in many fields. For example, in physics, understanding how graphs of motion change with different initial conditions or forces can be visualized through transformations. In computer graphics, transformations are used to manipulate objects in a scene, such as scaling, rotating, and translating them. In signal processing, transformations like time scaling and time reversal are used to analyze and manipulate signals.
The graph of the function y = − 4 x − 36 is stretched by a factor of 2, reflected over the y-axis, and translated 9 units to the left. Therefore, the correct answer is option D. The transformations include a vertical stretch, horizontal reflection, and leftward translation.
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