Detailed Examples of Graphing Square Root Functions with Multiple Transformations

📚 Quick Study Guide

• 🔍 The general form of a transformed square root function is $f(x) = a\sqrt{b(x - h)} + k$, where:
• 📈 $a$ represents a vertical stretch/compression and reflection across the x-axis if $a < 0$.
• ↔️ $b$ represents a horizontal stretch/compression and reflection across the y-axis if $b < 0$.
• ➡️ $h$ represents a horizontal translation (shift).
• ⬆️ $k$ represents a vertical translation (shift).
• 📝 The parent function is $f(x) = \sqrt{x}$, which starts at (0, 0) and increases.
• 💡 To graph, identify $a$, $b$, $h$, and $k$, then apply transformations in the correct order. A good order is stretches/compressions/reflections first, then translations.

🧪 Practice Quiz

1. What transformation does $f(x) = -\sqrt{x}$ represent compared to the parent function $f(x) = \sqrt{x}$?

   1. Vertical stretch by a factor of -1
   2. Horizontal shift to the left
   3. Reflection across the x-axis
   4. Reflection across the y-axis

2. What is the domain of the function $f(x) = \sqrt{x - 3}$?

   1. $x < 3$
   2. $x \le 3$
   3. $x > 3$
   4. $x \ge 3$

3. What is the range of the function $f(x) = 2\sqrt{x} + 1$?

   1. $y < 1$
   2. $y \le 1$
   3. $y > 1$
   4. $y \ge 1$

4. Which transformation shifts the graph of $f(x) = \sqrt{x}$ two units to the right?

   1. $f(x) = \sqrt{x} + 2$
   2. $f(x) = \sqrt{x} - 2$
   3. $f(x) = \sqrt{x + 2}$
   4. $f(x) = \sqrt{x - 2}$

5. What is the effect of the transformation in the function $f(x) = \sqrt{-x}$?

   1. Vertical shift
   2. Horizontal shift
   3. Reflection across the x-axis
   4. Reflection across the y-axis

6. How does the graph of $f(x) = \frac{1}{2}\sqrt{x}$ compare to the graph of $f(x) = \sqrt{x}$?

   1. Steeper
   2. Less steep
   3. Shifted upwards
   4. Shifted downwards

7. Which function represents a square root function reflected across the x-axis and shifted 3 units up?

   1. $f(x) = \sqrt{x} + 3$
   2. $f(x) = -\sqrt{x} - 3$
   3. $f(x) = -\sqrt{x} + 3$
   4. $f(x) = \sqrt{x} - 3$