Question

graph the equation shown below by transforming the given graph of the parent function.

$y = 2^{x}+3$

start over

Explanation:

Step1: Identify the parent function

The parent function of \(y = 2^{x}+3\) is \(y = 2^{x}\).

Step2: Recall the transformation rule

For a function \(y = f(x)+k\), if \(k>0\), the graph of \(y = f(x)\) is shifted up by \(k\) units. Here \(f(x)=2^{x}\) and \(k = 3\).

Step3: Transform key points

• For the parent function \(y = 2^{x}\), when \(x = 0\), \(y=2^{0}=1\); when \(x = 1\), \(y = 2^{1}=2\); when \(x = 2\), \(y=2^{2}=4\).
• For the function \(y = 2^{x}+3\), when \(x = 0\), \(y=2^{0}+3=1 + 3=4\); when \(x = 1\), \(y=2^{1}+3=2 + 3=5\); when \(x = 2\), \(y=2^{2}+3=4+3 = 7\).

Answer:

Shift the graph of \(y = 2^{x}\) (the given graph) up by \(3\) units. The key points \((0,1)\), \((1,2)\), \((2,4)\) of \(y = 2^{x}\) are transformed to \((0,4)\), \((1,5)\), \((2,7)\) for \(y = 2^{x}+3\).