Answer :
Let's analyze the given function [tex]\( f(x) = a^{(x+h)} + k \)[/tex] to determine which option is correct based on the behavior of the graph.
1. Understanding the Components of the Function:
- [tex]\( a \)[/tex]: The base of the exponential function.
- [tex]\( x \)[/tex]: The variable.
- [tex]\( h \)[/tex]: A horizontal shift of the graph.
- [tex]\( k \)[/tex]: A vertical shift of the graph.
2. Behavior of Exponential Functions:
- If [tex]\( a > 1 \)[/tex], the function [tex]\( a^{x+h} \)[/tex] tends to increase as [tex]\( x \)[/tex] increases. This is typical for exponential growth.
- If [tex]\( 0 < a < 1 \)[/tex], the function [tex]\( a^{x+h} \)[/tex] tends to decrease as [tex]\( x \)[/tex] increases. This is typical for exponential decay.
- If [tex]\( a < 0 \)[/tex], the base of the exponential function is negative, which is not usually considered in standard exponential function analysis because it can lead to undefined or complex values.
- The parameter [tex]\( k \)[/tex] vertically shifts the graph by [tex]\( k \)[/tex] units. If [tex]\( k < 1 \)[/tex], it simply shifts the entire graph downward by [tex]\( k \)[/tex] units.
3. Choosing the Correct Option:
- Option A: [tex]\( a > 1 \)[/tex] suggests the function is exponentially increasing.
- Option B: [tex]\( k < 1 \)[/tex] suggests a vertical downward shift, but does not influence the nature of the exponential growth or decay determined by [tex]\( a \)[/tex].
- Option C: [tex]\( a < 0 \)[/tex] is generally not considered for standard exponential functions.
- Option D: [tex]\( 0 < a < 1 \)[/tex] suggests the function is exponentially decreasing.
Given these analyses, the correct condition that matches the shape of the graph of [tex]\( f(x) = a^{(x+h)} + k \)[/tex] given the options is:
Option D: [tex]\( 0 < a < 1 \)[/tex]
1. Understanding the Components of the Function:
- [tex]\( a \)[/tex]: The base of the exponential function.
- [tex]\( x \)[/tex]: The variable.
- [tex]\( h \)[/tex]: A horizontal shift of the graph.
- [tex]\( k \)[/tex]: A vertical shift of the graph.
2. Behavior of Exponential Functions:
- If [tex]\( a > 1 \)[/tex], the function [tex]\( a^{x+h} \)[/tex] tends to increase as [tex]\( x \)[/tex] increases. This is typical for exponential growth.
- If [tex]\( 0 < a < 1 \)[/tex], the function [tex]\( a^{x+h} \)[/tex] tends to decrease as [tex]\( x \)[/tex] increases. This is typical for exponential decay.
- If [tex]\( a < 0 \)[/tex], the base of the exponential function is negative, which is not usually considered in standard exponential function analysis because it can lead to undefined or complex values.
- The parameter [tex]\( k \)[/tex] vertically shifts the graph by [tex]\( k \)[/tex] units. If [tex]\( k < 1 \)[/tex], it simply shifts the entire graph downward by [tex]\( k \)[/tex] units.
3. Choosing the Correct Option:
- Option A: [tex]\( a > 1 \)[/tex] suggests the function is exponentially increasing.
- Option B: [tex]\( k < 1 \)[/tex] suggests a vertical downward shift, but does not influence the nature of the exponential growth or decay determined by [tex]\( a \)[/tex].
- Option C: [tex]\( a < 0 \)[/tex] is generally not considered for standard exponential functions.
- Option D: [tex]\( 0 < a < 1 \)[/tex] suggests the function is exponentially decreasing.
Given these analyses, the correct condition that matches the shape of the graph of [tex]\( f(x) = a^{(x+h)} + k \)[/tex] given the options is:
Option D: [tex]\( 0 < a < 1 \)[/tex]
