Question

describe the transformations
a. (y = 2sqrt{x - 3} + 7)

Explanation:

⚡ Using what you learned: combining transformations

Step 1: Identify the parent function

The base function before any transformations is the square root function:

$$ y = \sqrt{x} $$

Step 2: Analyze the horizontal shift

The term inside the radical is \( x - 3 \). Replacing \( x \) with \( x - h \) shifts the graph horizontally:

$$ y = \sqrt{x - 3} $$

This represents a horizontal shift to the right by \( 3 \) units.

Step 3: Analyze the vertical stretch

The coefficient of \( 2 \) multiplied by the radical stretches the function vertically:

$$ y = 2\sqrt{x - 3} $$

This represents a vertical stretch by a factor of \( 2 \).

Step 4: Analyze the vertical shift

The constant \( +7 \) added to the outside of the function shifts the graph vertically:

$$ y = 2\sqrt{x - 3} + 7 $$

This represents a vertical shift up by \( 7 \) units.

Answer:

The transformations applied to the parent function \( y = \sqrt{x} \) to obtain \( y = 2\sqrt{x - 3} + 7 \) are:

1. Horizontal shift to the right by \( 3 \) units.
2. Vertical stretch by a factor of \( 2 \).
3. Vertical shift up by \( 7 \) units.