Answer :
To find the equation of the graph after it is translated, we need to understand how translations affect the equation of a graph. Let's go through this step-by-step.
### Step 1: Translation to the Left
When a graph is translated horizontally to the left by a certain number of units, this affects the \( x \)-values in the equation. Specifically, if we want to translate the graph 6 units to the left, we replace \( x \) with \( x + 6 \).
Initially, our equation is:
[tex]\[ y = 7^x \][/tex]
After translating it 6 units to the left, the x in the exponent becomes \( x + 6 \):
[tex]\[ y = 7^{x+6} \][/tex]
### Step 2: Translation Upwards
When a graph is translated vertically upwards by a certain number of units, this affects the \( y \)-values in the equation. Specifically, if we want to translate the graph 3 units upward, we add 3 to the entire function.
From the previous step, our equation is:
[tex]\[ y = 7^{x+6} \][/tex]
After translating it 3 units upward, we add 3:
[tex]\[ y = 7^{x+6} + 3 \][/tex]
### Conclusion
Putting both steps together, the complete translation involves shifting the graph 6 units to the left and then 3 units upward. The new equation of the graph is:
[tex]\[ y = 7^{x+6} + 3 \][/tex]
### Choosing the Correct Option
Among the given choices, this corresponds to:
B. \( y = 7^{x+6} + 3 \)
Therefore, the correct equation of the graph in its final position is:
[tex]\[ y = 7^{x+6} + 3 \][/tex]
And the correct option is:
B. [tex]\( y = 7^{x+6} + 3 \)[/tex]
### Step 1: Translation to the Left
When a graph is translated horizontally to the left by a certain number of units, this affects the \( x \)-values in the equation. Specifically, if we want to translate the graph 6 units to the left, we replace \( x \) with \( x + 6 \).
Initially, our equation is:
[tex]\[ y = 7^x \][/tex]
After translating it 6 units to the left, the x in the exponent becomes \( x + 6 \):
[tex]\[ y = 7^{x+6} \][/tex]
### Step 2: Translation Upwards
When a graph is translated vertically upwards by a certain number of units, this affects the \( y \)-values in the equation. Specifically, if we want to translate the graph 3 units upward, we add 3 to the entire function.
From the previous step, our equation is:
[tex]\[ y = 7^{x+6} \][/tex]
After translating it 3 units upward, we add 3:
[tex]\[ y = 7^{x+6} + 3 \][/tex]
### Conclusion
Putting both steps together, the complete translation involves shifting the graph 6 units to the left and then 3 units upward. The new equation of the graph is:
[tex]\[ y = 7^{x+6} + 3 \][/tex]
### Choosing the Correct Option
Among the given choices, this corresponds to:
B. \( y = 7^{x+6} + 3 \)
Therefore, the correct equation of the graph in its final position is:
[tex]\[ y = 7^{x+6} + 3 \][/tex]
And the correct option is:
B. [tex]\( y = 7^{x+6} + 3 \)[/tex]
To determine the equation of the graph \( y = 7^x \) after it is translated 6 units to the left and 3 units upward, follow these steps:
1. **Horizontal Translation**: Translating 6 units to the left replaces \( x \) with \( x + 6 \):
\[
y = 7^{x + 6}
\]
2. **Vertical Translation**: Translating 3 units upward adds 3 to the entire function:
\[
y = 7^{x + 6} + 3
\]
Thus, the final equation of the graph is:
\[
y = 7^{x + 6} + 3
\]
The correct answer is:
\[
\boxed{y = 7^{x + 6} + 3}
\]
