Question

how does the graph of ( y = f(x + 2) ) differ from the graph of ( y = f(x - 3) )?
the graph of ( y = f(x + 2) ) is units the graph of ( y = f(x - 3) ).

Explanation:

Step1: Recall the horizontal shift rule

For a function \(y = f(x - h)\), if \(h>0\), the graph shifts \(h\) units to the right. For \(y = f(x + k)\) (\(k>0\)), the graph shifts \(k\) units to the left.
For \(y = f(x + 2)=f(x-(- 2))\), the graph of \(y = f(x)\) shifts \(2\) units to the left. For \(y = f(x - 3)\), the graph of \(y = f(x)\) shifts \(3\) units to the right.

Step2: Calculate the horizontal distance between the two - shifted graphs

The distance between the two shifted graphs is \(2+3=5\) units. And since the left - shifted graph (\(y = f(x + 2)\)) is to the left of the right - shifted graph (\(y = f(x - 3)\))

Answer:

The graph of \(y = f(x + 2)\) is \(5\) units to the left of the graph of \(y = f(x - 3)\)