Shifting Functions Math Example 4

Follow the full solution, then compare it with the other examples linked below.

medium

Write the equation of the function whose graph is

y=\sqrt{x}

shifted right

9

and reflected over the

x

-axis. State the domain.

1
Shift right $9$ : $\sqrt{x-9}$ . Reflect over $x$ -axis (multiply by $-1$ ): $-\sqrt{x-9}$ .
2
Domain: $x-9\geq0 \Rightarrow x\geq9$ . Domain $=[9,\infty)$ .

Answer

y=-\sqrt{x-9}

; domain

[9,\infty)

Combining a shift with a reflection: first shift the argument (

x\to x-9

), then apply the vertical reflection (multiply by

-1

). The domain shifts with the function — the square root requires its argument to be non-negative.

About Shifting Functions

Shifting a function translates its graph horizontally or vertically without changing its shape: $f(x - h) + k$ shifts right by $h$ and up by $k$ .

Starting from [formula], describe and sketch the transformations for [formula].

The graph of [formula] is shifted left [formula] units and down [formula] units to give [formula]. W

The point [formula] is on the graph of [formula]. Find the corresponding point on each shifted graph