Question

the graph of $y = f(x)$ is the solid black graph below. which function represents the dotted graph?
answer
$\circ\\ y = f(x - 2) - 2$ $\circ\\ y = f(x + 2) + 2$
$\circ\\ y = f(x - 2) + 2$ $\circ\\ y = f(x + 2) - 2$

Explanation:

Step1: Analyze horizontal shift

The solid graph \( y = f(x) \) and the dotted graph. Let's take a key point, like the vertex of the absolute - value - like graph (the origin for the solid graph). For the dotted graph, the vertex seems to be at \( (- 2,-2) \) relative to the solid graph's vertex \((0,0)\).

For horizontal shifts: If we have a function \( y=f(x + h) \), when \( h>0 \), the graph shifts left by \( h \) units; when \( h < 0 \), it shifts right by \(|h|\) units. To get from \( x = 0 \) (solid vertex) to \( x=-2 \) (dotted vertex), we need to shift left by 2 units. So the horizontal shift is \( x\to x + 2 \), which means the function has the form \( y = f(x + 2)\) (before considering vertical shift).

Step2: Analyze vertical shift

For vertical shifts: If we have a function \( y=f(x)+k \), when \( k>0 \), the graph shifts up by \( k \) units; when \( k < 0 \), it shifts down by \(|k|\) units. To get from \( y = 0 \) (solid vertex) to \( y=-2 \) (dotted vertex), we need to shift down by 2 units. So the vertical shift is \( k=-2 \), which means the function has the form \( y=f(x + 2)-2 \).

Answer:

\( y = f(x + 2)-2 \)