Answer :
To determine how the function [tex]\( f(x) = 2^{x+4} \)[/tex] shifts five units to the right, and to find the corresponding value of [tex]\( k \)[/tex] for the new function [tex]\( f(x) = 2^{x-k} \)[/tex], follow these steps:
1. Understanding the Shift:
- When a graph of a function shifts to the right by a certain number of units, [tex]\( x \)[/tex] in the function is effectively replaced by [tex]\( x - \text{shift} \)[/tex]. For a shift to the right by 5 units, we replace [tex]\( x \)[/tex] with [tex]\( x - 5 \)[/tex].
2. Create the New Function:
- Starting with the original function [tex]\( f(x) = 2^{x+4} \)[/tex], we apply the shift:
[tex]\[ x \rightarrow x - 5 \][/tex]
- So the new function becomes:
[tex]\[ f(x - 5) = 2^{(x - 5) + 4} \][/tex]
3. Simplify the New Function:
- Simplify the exponent in the new function:
[tex]\[ 2^{(x - 5) + 4} = 2^{x - 1} \][/tex]
4. Compare to the Desired Form:
- The new function after the shift is [tex]\( 2^{x-1} \)[/tex]. We need this function in the form [tex]\( 2^{x - k} \)[/tex] to determine [tex]\( k \)[/tex].
5. Conclude the Value of [tex]\( k \)[/tex]:
- By comparing [tex]\( 2^{x - 1} \)[/tex] with [tex]\( 2^{x - k} \)[/tex], we see that [tex]\( k = 1 \)[/tex].
Therefore, the value of [tex]\( k \)[/tex] is [tex]\( \boxed{1} \)[/tex].
1. Understanding the Shift:
- When a graph of a function shifts to the right by a certain number of units, [tex]\( x \)[/tex] in the function is effectively replaced by [tex]\( x - \text{shift} \)[/tex]. For a shift to the right by 5 units, we replace [tex]\( x \)[/tex] with [tex]\( x - 5 \)[/tex].
2. Create the New Function:
- Starting with the original function [tex]\( f(x) = 2^{x+4} \)[/tex], we apply the shift:
[tex]\[ x \rightarrow x - 5 \][/tex]
- So the new function becomes:
[tex]\[ f(x - 5) = 2^{(x - 5) + 4} \][/tex]
3. Simplify the New Function:
- Simplify the exponent in the new function:
[tex]\[ 2^{(x - 5) + 4} = 2^{x - 1} \][/tex]
4. Compare to the Desired Form:
- The new function after the shift is [tex]\( 2^{x-1} \)[/tex]. We need this function in the form [tex]\( 2^{x - k} \)[/tex] to determine [tex]\( k \)[/tex].
5. Conclude the Value of [tex]\( k \)[/tex]:
- By comparing [tex]\( 2^{x - 1} \)[/tex] with [tex]\( 2^{x - k} \)[/tex], we see that [tex]\( k = 1 \)[/tex].
Therefore, the value of [tex]\( k \)[/tex] is [tex]\( \boxed{1} \)[/tex].
