Describe the transformations from the parent function.

Name: Unit 6: Radical Functions Homework

Date:

Bell:

This is a 2-page document.

For questions 1-2: Describe the transformations from the parent function.

S(x) = -2x – 9
f(x) = x + 5 + 3
The square root parent function is vertically compressed by a factor of 1/3, then translated so that it has an endpoint located at (4, -1). Write an equation that could represent this function.
The cube root parent function is reflected across the x-axis, vertically stretched by a factor of 3 then translated 8 units down. Write an equation that could represent this function.

The Correct Answer and Explanation is:

---

Question 2:

Given:
f(x)=−x+53+3f(x) = -\sqrt[3]{x+5} + 3

Transformations from the parent function f(x)=x3f(x) = \sqrt[3]{x}:

1. Inside the root: x+5x + 5
   → This is a horizontal shift 5 units to the left.
2. Negative sign in front: −x+53-\sqrt[3]{x+5}
   → This is a reflection across the x-axis.
3. Outside the root: +3+3
   → This is a vertical shift 3 units up.

✅ Final answer for #2:

• Shift left 5 units
• Reflect across the x-axis
• Shift up 3 units

---

Question 4:

We are asked to write an equation for the cube root parent function, given these transformations:

• Reflected across the x-axis
• Vertically stretched by a factor of 3
• Translated 8 units down

Start with the parent function:
f(x)=x3f(x) = \sqrt[3]{x}

Apply transformations:

1. Reflection across x-axis → Multiply by -1:
   f(x)=−x3f(x) = -\sqrt[3]{x}
2. Vertical stretch by 3 → Multiply by 3:
   f(x)=−3x3f(x) = -3\sqrt[3]{x}
3. Translate 8 units down → Subtract 8:
   f(x)=−3×3−8f(x) = -3\sqrt[3]{x} – 8

✅ Final equation for #4:
f(x)=−3×3−8f(x) = -3\sqrt[3]{x} – 8

---

Explanation

Transformations of functions involve shifting, stretching, compressing, or reflecting the graph of a parent function. Understanding how each change affects the graph helps us write or interpret equations correctly.

For question #2, we start with the cube root parent function f(x)=x3f(x) = \sqrt[3]{x}. The term x+5x + 5 inside the radical shifts the graph left by 5 units, because the inside of the function affects the horizontal position in the opposite direction. The negative sign in front reflects the graph over the x-axis, flipping it upside down. Finally, adding 3 outside the function shifts the graph up by 3 units, moving every point vertically.

For question #4, we apply a sequence of transformations starting with the parent cube root function. Reflecting over the x-axis changes the sign of the function, so x3\sqrt[3]{x} becomes −x3-\sqrt[3]{x}. Then, stretching the function vertically by a factor of 3 means that all y-values are three times farther from the x-axis, changing it to −3×3-3\sqrt[3]{x}. Lastly, subtracting 8 translates the function downward by 8 units. So the full equation becomes f(x)=−3×3−8f(x) = -3\sqrt[3]{x} – 8.

These types of transformations help students understand how different algebraic modifications affect a function’s graph and behavior.