Question

the graph of $y = f(x)$ is the solid black graph below. which function represents the dotted graph?
answer
$\circ$ $y = f(-x) + 3$
$\circ$ $y = -f(x + 3)$
$\circ$ $y = f(-x) - 3$
$\circ$ $y = -f(x - 3)$

Explanation:

1. First, analyze the transformation of the graph. The solid graph is \( y = f(x) \), and the dotted graph seems to be a reflection over the y - axis (which corresponds to replacing \( x \) with \( -x \)) and a reflection over the x - axis (which corresponds to multiplying the function by - 1) and a horizontal shift? Wait, no, let's check the options.

• Option \( y=-f(x + 3) \): This is a horizontal shift left by 3 units and a reflection over the x - axis.
• Option \( y = f(-x)+3 \): Reflection over y - axis and vertical shift up 3.
• Option \( y = f(-x)-3 \): Reflection over y - axis and vertical shift down 3.
• Option \( y=-f(x - 3) \): Horizontal shift right by 3 units and reflection over x - axis.
• Looking at the graph, the dotted graph is a reflection over the y - axis (so \( x\to - x \)) and a reflection over the x - axis (multiply by - 1) and maybe a shift? Wait, no, let's re - examine. Wait, the original function \( y = f(x) \) (solid) is a V - shape with vertex at (0,0). The dotted graph: let's see the symmetry. If we reflect \( f(x) \) over the y - axis, we get \( f(-x) \), but the dotted graph is also reflected over the x - axis? Wait, no, maybe I made a mistake. Wait, the original function \( y = f(x) \) (solid) has positive y for positive x and negative y for negative x? Wait, no, the solid graph: when \( x>0 \), \( y>0 \); when \( x < 0 \), \( y < 0 \). The dotted graph: when \( x<0 \), \( y>0 \); when \( x>0 \), \( y < 0 \)? Wait, no, the dotted graph is in the second and third quadrants? Wait, maybe the correct transformation is \( y=-f(x + 3) \)? No, wait, let's check the vertex. The original vertex is (0,0). The dotted graph's vertex: let's see the grid. If we consider \( y=-f(x + 3) \), the vertex would be at (- 3,0), and the graph is a reflection over the x - axis and shifted left by 3. But if we look at the dotted graph, it seems to be a reflection over the y - axis and a reflection over the x - axis? Wait, no, maybe the correct option is \( y=-f(x + 3) \)? Wait, no, let's think again. Wait, the original function \( f(x) \): for \( x\geq0 \), \( y = x \); for \( x<0 \), \( y=-x \) (since it's a V - shape with slope 1 for \( x\geq0 \) and slope - 1 for \( x < 0 \)). Then \( -f(x + 3) \): for \( x+3\geq0\) (i.e., \( x\geq - 3 \)), \( y=-(x + 3)=-x - 3 \); for \( x + 3<0\) (i.e., \( x<-3 \)), \( y=-(-(x + 3))=x + 3 \). Which matches the dotted graph's shape (a V - shape opening down, shifted left by 3). The other options: \( f(-x) \) would reflect over y - axis, making \( y = -x \) for \( x\leq0 \) and \( y = x \) for \( x>0 \), which is not the dotted graph. \( f(-x)+3 \) would shift up, \( f(-x)-3 \) shift down, \( -f(x - 3) \) shift right. So the correct option is \( y=-f(x + 3) \).

Answer:

\( \boldsymbol{y=-f(x + 3)} \) (the option corresponding to \( y=-f(x + 3) \))