Question

graph
\\y = -\frac{1}{4}\sqrt{x + 2} + 1\\

Explanation:

⚡ Using: combining transformations · plotting points and graphing figures

Step 1: Identify the starting point (vertex)

The parent function is \( f(x) = \sqrt{x} \), which starts at \( (0,0) \).

For the transformed function:

$$ y = -\frac{1}{4}\sqrt{x + 2} + 1 $$

• Horizontal shift: The term \( x + 2 \) inside the radical shifts the graph left by \( 2 \) units.
• Vertical shift: The \( +1 \) at the end shifts the graph up by \( 1 \) unit.

Thus, the starting point of the curve is:

$$ (-2, 1) $$

Step 2: Determine the shape and direction

• Vertical reflection: The negative sign in front of the radical, \( -\frac{1}{4} \), reflects the graph vertically across the \( x \)-axis. Instead of curving upwards, the graph curves downwards.
• Vertical compression: The factor of \( \frac{1}{4} \) compresses the graph vertically, making it flatter than the standard square root curve.

Step 3: Calculate key points for plotting

Choose values of \( x \) that make the expression under the radical, \( x + 2 \), a perfect square (\( 0, 1, 4, 16 \)):

• For \( x = -2 \):

$$ y = -\frac{1}{4}\sqrt{-2 + 2} + 1 = 1 \implies (-2, 1) $$

• For \( x = -1 \):

$$ y = -\frac{1}{4}\sqrt{-1 + 2} + 1 = -\frac{1}{4}(1) + 1 = 0.75 \implies (-1, 0.75) $$

• For \( x = 2 \):

$$ y = -\frac{1}{4}\sqrt{2 + 2} + 1 = -\frac{1}{4}(2) + 1 = 0.5 \implies (2, 0.5) $$

• For \( x = 14 \):

$$ y = -\frac{1}{4}\sqrt{14 + 2} + 1 = -\frac{1}{4}(4) + 1 = 0 \implies (14, 0) $$

Step 4: Sketch the graph

1. Plot the starting point at \( (-2, 1) \).
2. Plot the additional points: \( (-1, 0.75) \), \( (2, 0.5) \), and the \( x \)-intercept at \( (14, 0) \).
3. Draw a smooth curve starting from \( (-2, 1) \) that extends to the right, gradually curving downwards through the plotted points.

Answer:

To graph the function \( y = -\frac{1}{4}\sqrt{x + 2} + 1 \):

1. Starting Point: \( (-2, 1) \)
2. Direction: Curves to the right and downwards.
3. Key Points to Plot:

• \( (-2, 1) \)
• \( (-1, 0.75) \)
• \( (2, 0.5) \)
• \( (14, 0) \)