Question

the graph of ( f(x)=sqrt{x} ) is shown on the grid. which statements about the relationship between the graph of ( f(x) ) and the graph of ( g(x)=sqrt{-9(x + 2)}-7 ) are true? select all that apply.

Explanation:

Step1: Rewrite $g(x)$ in standard form

First, rewrite the radicand to match the transformation form of $f(x)=\sqrt{x}$:
$$g(x) = \sqrt{-9(x+2)} - 7 = 3\sqrt{-(x+2)} - 7$$

Step2: Analyze horizontal transformations

For $f(x)=\sqrt{x}$, replacing $x$ with $x+2$ shifts the graph 2 units left. Replacing $x$ with $-(x+2)$ reflects the graph across the y-axis.

Step3: Analyze vertical transformations

Multiplying the radical by 3 vertically stretches the graph by a factor of 3. Subtracting 7 shifts the graph 7 units down.

Step4: Verify domain shift

Domain of $f(x)$: $x\geq0$. Domain of $g(x)$: solve $-9(x+2)\geq0 \implies x+2\leq0 \implies x\leq-2$, which matches the left shift and reflection.

Answer:

The true statements (based on radical function transformations) are:

1. The graph of $g(x)$ is the graph of $f(x)$ reflected across the y-axis.
2. The graph of $g(x)$ is the graph of $f(x)$ shifted 2 units to the left.
3. The graph of $g(x)$ is the graph of $f(x)$ vertically stretched by a factor of 3.
4. The graph of $g(x)$ is the graph of $f(x)$ shifted 7 units downward.