6. given the following graph of the function f(x) below. sketch the following graph g(x)=f(x - 4)

Explanation:

Step1: Recall transformation rule

The transformation $y = f(x - h)$ is a horizontal shift of the graph of $y=f(x)$. If $h>0$, the graph shifts to the right by $h$ units. Here $h = 4$, so the graph of $f(x)$ shifts 4 units to the right.

Step2: Shift key - points

For the graph of $y = f(x)$, find key - points such as the x - intercepts, vertices (if it's a non - linear function like a parabola or an absolute - value - like function). Let's assume some key - points on $y = f(x)$ are $(x_1,y_1),(x_2,y_2),\cdots$. For the graph of $y = f(x - 4)$, the new key - points will be $(x_1 + 4,y_1),(x_2+4,y_2),\cdots$. For example, if the vertex of $f(x)$ is at $(0,0)$, the vertex of $g(x)=f(x - 4)$ will be at $(4,0)$. If an x - intercept of $f(x)$ is at $x=-5$, the corresponding x - intercept of $g(x)$ will be at $x=-5 + 4=-1$.

Step3: Sketch the new graph

Using the shifted key - points, draw the same shape as the graph of $f(x)$. The overall shape of the graph remains the same, just its position on the x - axis changes.

Answer:

Sketch the graph of $f(x)$ shifted 4 units to the right. The shape of the graph is identical to that of $f(x)$, but all points on $f(x)$ are moved 4 units to the right along the x - axis.