Question

below is the graph of $y = 2^x$. translate it to become the graph of $y = 2^{x + 3}+1$.

Explanation:

Step1: Horizontal translation

For the function \(y = f(x)\) to \(y=f(x + h)\), if \(h>0\), the graph shifts left by \(h\) units.
For \(y = 2^{x}\) to \(y = 2^{x+3}=2^{(x - (- 3))}\), we shift the graph of \(y = 2^{x}\) left by \(3\) units.

Step2: Vertical translation

For the function \(y = f(x)\) to \(y=f(x)+k\), if \(k > 0\), the graph shifts up by \(k\) units.
After shifting \(y = 2^{x}\) left by \(3\) units to get \(y = 2^{x + 3}\), for \(y=2^{x+3}+1\), we shift the graph of \(y = 2^{x+3}\) up by \(1\) unit.

Answer:

Shift the graph of \(y = 2^{x}\) left by \(3\) units and then up by \(1\) unit.