Question

describe the transformation \\(f(x) = \sqrt{x+1} - 5\\) in words.

the parent function \\(f(x) = \sqrt{x}\\) has been shifted left 1 unit and down 5 units.

identify the graph.

Explanation:

⚡ Using what you learned: combining transformations

Step 1: Analyze the horizontal shift

The parent function is \( f(x) = \sqrt{x} \).
In the transformed function \( f(x) = \sqrt{x + 1} - 5 \), the term inside the radical is \( x + 1 \).

Adding a positive constant inside the function's argument shifts the graph horizontally to the left:

$$ x + 1 = 0 \implies x = -1 $$

Thus, the graph is shifted left by \( 1 \) unit.

Step 2: Analyze the vertical shift

The constant term outside the radical is \( -5 \).
Subtracting a constant outside the function shifts the graph vertically downward:

$$ y = -5 $$

Thus, the graph is shifted down by \( 5 \) units.

Step 3: Identify the correct graph

The starting point (vertex) of the parent square root function \( f(x) = \sqrt{x} \) is at \( (0,0) \).
Applying the shifts:

• Shift left by \( 1 \) unit: \( x = -1 \)
• Shift down by \( 5 \) units: \( y = -5 \)

The new starting point of the curve must be at \( (-1, -5) \).

Looking at the given options:

• Graph 1: A parabola (quadratic function).
• Graph 2: A cubic-like curve.
• Graph 3: A square root curve starting at \( (-1, -5) \).
• Graph 4: A straight line (linear function).

The correct graph is the third graph.

Answer:

• First blank: left
• Second blank: down
• Correct Graph: The third graph (the square root curve starting at \( (-1, -5) \))