6. use transformations to sketch the graph of the following function.\n\n$$f(x) = \\sqrt{x + 4} - 3$$

Explanation:

Step1: Identify the parent function

The parent function is the square root function: $$y = \sqrt{x}$$

Step2: Determine the horizontal shift

The term $(x + 4)$ shifts the graph left by 4 units: $$y = \sqrt{x + 4}$$

Step3: Determine the vertical shift

The constant $-3$ shifts the graph down by 3 units: $$f(x) = \sqrt{x + 4} - 3$$

Step4: Identify key points for sketching

The original starting point $(0,0)$ moves to $(-4, -3)$. Another point $(1,1)$ moves to $(-3, -2)$.

Answer:

To sketch the graph of $f(x) = \sqrt{x + 4} - 3$, start with the parent graph $y = \sqrt{x}$, shift it horizontally 4 units to the left, and then shift it vertically 3 units down. The resulting graph starts at the point $(-4, -3)$ and passes through points such as $(-3, -2)$ and $(0, -1)$.