Question

exponential functions
drag all of the exponential functions to the corresponding translations.

exponential function translations
______________________ right 2, up 1
______________________ left 2, up 1
______________________ left 2, down 1
______________________ right 2, down 1

y=3^{x+2}+1
y=3^{x-2}-1
y=3^{x-2}+1
y=3^{x+2}-1

Explanation:

Answer:

To solve this, we use the rules for horizontal and vertical translations of exponential functions. For a function \( y = a^{x - h}+k \):

• Horizontal translation: \( h \) units to the right if \( h>0 \), \( |h| \) units to the left if \( h < 0 \).
• Vertical translation: \( k \) units up if \( k>0 \), \( |k| \) units down if \( k < 0 \).

Step 1: Analyze \( y = 3^{x + 2}+1 \)

Rewrite as \( y=3^{x-(-2)} + 1 \). Here, \( h=- 2\) (so left 2 units) and \( k = 1\) (up 1 unit). So it matches "left 2, up 1".

Step 2: Analyze \( y = 3^{x+2}-1 \)

Rewrite as \( y = 3^{x-(-2)}-1 \). \( h=-2\) (left 2 units) and \( k=-1\) (down 1 unit). So it matches "left 2, down 1".

Step 3: Analyze \( y = 3^{x - 2}-1 \)

Here, \( h = 2\) (right 2 units) and \( k=-1\) (down 1 unit). So it matches "right 2, down 1".

Step 4: Analyze \( y = 3^{x-2}+1 \)

Here, \( h = 2\) (right 2 units) and \( k = 1\) (up 1 unit). So it matches "right 2, up 1".

So the pairings are:

• \( y = 3^{x + 2}+1 \) → left 2, up 1
• \( y = 3^{x+2}-1 \) → left 2, down 1
• \( y = 3^{x - 2}-1 \) → right 2, down 1
• \( y = 3^{x-2}+1 \) → right 2, up 1