Question

the following transformations of $f(x)$ does the solid line represent?
a horizontal shift right 3 units and a vertical shift down 1 unit
a horizontal shift left 3 units and a vertical shift down 1 unit
a horizontal shift right 3 units and a reflection over the x-axis

Explanation:

1. Reflection Check: The dashed function $f(x)$ opens downward, while the solid line opens upward. This means the solid line is a reflection of $f(x)$ over the x-axis.
2. Horizontal Shift Check: The vertex of $f(x)$ is at $(0,0)$. The vertex of the solid line is at $(3,0)$, which is 3 units to the right of $f(x)$'s vertex.
3. Vertical Shift Check: Both vertices lie on the x-axis, so there is no vertical shift.
4. Match to Options: Only the third option describes a horizontal shift right 3 units and a reflection over the x-axis, which matches the observed transformations.

Answer:

A horizontal shift right 3 units and a reflection over the x-axis