Question

use the graph of ( y = f(x) ) to graph the function ( g(x)=f(x)+3 ).

Explanation:

Step1: Understand the transformation

The function \(g(x)=f(x)+3\) is a vertical shift of the function \(y = f(x)\). For any function \(y = f(x)\), the transformation \(y=f(x)+k\) shifts the graph of \(y = f(x)\) vertically. If \(k>0\), the shift is upwards. Here \(k = 3\), so we shift the graph of \(y=f(x)\) 3 units up.

Step2: Apply the transformation to key points

Let's consider some key points on \(y = f(x)\). Suppose a point \((x_0,y_0)\) lies on \(y = f(x)\). Then the corresponding point on \(y = g(x)\) is \((x_0,y_0 + 3)\).
For example, if on \(y=f(x)\) we have a point \((-4,0)\) (assuming from the visual - approximate), then on \(y = g(x)\) the point is \((-4,0 + 3)=(-4,3)\). If we have a point \((2,-2)\) on \(y=f(x)\), then on \(y = g(x)\) the point is \((2,-2+3)=(2,1)\). If we have a point \((4,0)\) on \(y=f(x)\), then on \(y = g(x)\) the point is \((4,0 + 3)=(4,3)\)

Answer:

To graph \(g(x)=f(x)+3\), take each point \((x,y)\) on the graph of \(y = f(x)\) and move it 3 units up to the point \((x,y + 3)\) and then connect the new - transformed points.