Let’s analyze the transformation of the graph of the function [tex]\( f(x) = 2^x \)[/tex] to the graph of the function [tex]\( f(x) = -3 \cdot 2^x \)[/tex].
### Step-by-Step Analysis:
1. Compare the Basic Functions:
- [tex]\( f(x) = 2^x \)[/tex]: This is an exponential growth function because as [tex]\( x \)[/tex] increases, [tex]\( 2^x \)[/tex] increases exponentially.
- [tex]\( f(x) = -3 \cdot 2^x \)[/tex]: This function modifies [tex]\( f(x) = 2^x \)[/tex] in two ways - by multiplying it by -3.
2. Effect of the Coefficient -3:
Negative Sign:
- [tex]\( -2^x \)[/tex]: The negative sign reflects the graph of [tex]\( 2^x \)[/tex] over the [tex]\( x \)[/tex]-axis. This means that for every point [tex]\( (x, y) \)[/tex] on the graph of [tex]\( 2^x \)[/tex], the corresponding point on the graph of [tex]\( -2^x \)[/tex] will be [tex]\( (x, -y) \)[/tex].
Factor of 3:
- [tex]\( 3 \cdot 2^x \)[/tex]: Multiplying the function [tex]\( 2^x \)[/tex] by 3 causes a vertical stretch by a factor of 3. This means that if [tex]\( (x, y) \)[/tex] is a point on the graph of [tex]\( 2^x \)[/tex], then [tex]\( (x, 3y) \)[/tex] is a point on the graph of [tex]\( 3 \cdot 2^x \)[/tex].
3. Combined Effect:
- Combining these effects, we get [tex]\( f(x) = -3 \cdot 2^x \)[/tex]. This function reflects the graph over the [tex]\( x \)[/tex]-axis and then stretches it vertically by a factor of 3.
4. Determine the Correct Descriptions:
Exponential Decay: The function [tex]\( f(x) = -3 \cdot 2^x \)[/tex] will exhibit exponential decay after the reflection, but strictly speaking, the term "exponential decay" applies to functions like [tex]\( f(x) = -a^x \)[/tex] where [tex]\( 0 < a < 1 \)[/tex]. Here, the main idea is the reflection transformation which dominates the change.
Reflected over the x-axis: Absolutely true, since the negative coefficient causes a reflection over the [tex]\( x \)[/tex]-axis.
Reflected over the y-axis: This would apply if the transformation were [tex]\( 2^{-x} \)[/tex], not relevant here.
Vertical Compression: Compression would occur if the absolute value of the multiplier was less than 1. Here, it is greater than 1 and thus implies a stretch.
Exponential Growth: No, the graph is reflected and stretched, causing it to move downwards rather than grow upwards.
Vertical Stretch: Multiplying by the coefficient [tex]\( 3 \)[/tex] (absolute value) scales the graph vertically. This is true.
### Conclusion:
The correct descriptions for the transformation of [tex]\( f(x) = 2^x \)[/tex] to [tex]\( f(x) = -3 \cdot 2^x \)[/tex] are:
1. Exponential decay
2. Reflected over the [tex]\( x \)[/tex]-axis
3. Vertical stretch
