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Which of the following describes the transformations of [tex]\( g(x) = -(2)^{x+4} - 2 \)[/tex] from the parent function [tex]\( f(x) = 2^x \)[/tex]?

A. Shift 4 units left, reflect over the [tex]\( x \)[/tex]-axis, shift 2 units down

B. Shift 4 units left, reflect over the [tex]\( y \)[/tex]-axis, shift 2 units down

C. Shift 4 units right, reflect over the [tex]\( x \)[/tex]-axis, shift 2 units down

D. Shift 4 units right, reflect over the [tex]\( y \)[/tex]-axis, shift 2 units down

Sagot :

GinnyAnswer
To determine the transformations that have been applied to the parent function [tex]\( f(x) = 2^x \)[/tex] to obtain the function [tex]\( g(x) = -(2)^{x+4} - 2 \)[/tex], we need to analyze the changes step by step.

1. Horizontal Shift:
- The expression [tex]\( x + 4 \)[/tex] indicates a horizontal shift. Specifically, adding 4 inside the function argument (i.e., as [tex]\( x + 4 \)[/tex]) shifts the graph to the left by 4 units.

2. Reflection:
- The negative sign in front of the function [tex]\( -(2)^{x+4} \)[/tex] denotes a reflection. Since the negative sign is outside the exponent, it reflects the graph over the [tex]\( x \)[/tex]-axis.

3. Vertical Shift:
- The term [tex]\( -2 \)[/tex] at the end of the function [tex]\( -(2)^{x+4} - 2 \)[/tex] indicates a vertical shift. Subtracting 2 from the function shifts the graph down by 2 units.

Taking all these transformations into account, [tex]\( g(x) = -(2)^{x+4} - 2 \)[/tex] is obtained from [tex]\( f(x) = 2^x \)[/tex] by:

- Shifting the graph 4 units to the left,
- Reflecting the graph over the [tex]\( x \)[/tex]-axis,
- Shifting the graph 2 units downward.

Therefore, the correct description of the transformations is:

Shift 4 units left, reflect over the [tex]\( x \)[/tex]-axis, shift 2 units down.
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