Radical Functions Date: Bell: Homework 7: Graphing Radical Functions This is a 2-page document. ** For questions 1-2: Describe the transformations from the parent function f(x) = √x. 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 2 units down. Write an equation that could represent this function.
The Correct Answer and Explanation is:
Let’s go through each question and provide correct answers with a clear explanation of the transformations:
1. f(x)=2x−9f(x) = 2\sqrt{x} – 9
Transformations from the parent function f(x)=xf(x) = \sqrt{x}:
- Vertical stretch by a factor of 2 (multiplies the output by 2).
- Vertical translation down 9 units (subtracting 9 from the output).
2. f(x)=−x+53+3f(x) = -\sqrt[3]{x + 5} + 3
Transformations from the parent function f(x)=x3f(x) = \sqrt[3]{x}:
- Horizontal shift left 5 units (inside the radical: x+5x + 5).
- Reflection over the x-axis (the negative sign in front).
- Vertical shift up 3 units (adding 3 to the entire function).
3. The square root function is compressed vertically by 13\frac{1}{3}, and endpoint at (4, -1):
The general square root function is:
f(x)=ax−h+kf(x) = a\sqrt{x – h} + k
Where:
- a=13a = \frac{1}{3} (vertical compression),
- (h,k)=(4,−1)(h, k) = (4, -1) (endpoint of transformation).
So the equation is:
f(x)=13x−4−1f(x) = \frac{1}{3} \sqrt{x – 4} – 1
4. The cube root function is reflected over the x-axis, vertically stretched by 3, and translated 2 units down:
The general cube root function is:
f(x)=ax−h3+kf(x) = a\sqrt[3]{x – h} + k
Given:
- Reflected over x-axis → negative sign,
- Stretched by 3 → multiply by 3,
- Translated down 2 → subtract 2.
So:
f(x)=−3×3−2f(x) = -3\sqrt[3]{x} – 2
Explanation:
This worksheet focuses on identifying and writing transformations of radical functions from their parent functions. The parent functions here are x\sqrt{x} and x3\sqrt[3]{x}, representing square root and cube root graphs, respectively.
In question 1, the equation f(x)=2x−9f(x) = 2\sqrt{x} – 9 shows a vertical stretch by 2 and a shift downward by 9. This means that every point on the graph is stretched vertically (y-values double) and then moved down.
In question 2, f(x)=−x+53+3f(x) = -\sqrt[3]{x + 5} + 3 applies several transformations: first, a left shift by 5 units because of x+5x + 5, then a reflection over the x-axis (the negative sign flips the graph upside down), followed by a shift upward by 3 units.
Question 3 is more involved. We are told the function is a compressed version of x\sqrt{x} (multiplied by 13\frac{1}{3}), and it’s translated to end at the point (4, -1). Using the general form of a square root function with transformations, we substitute the point (4, -1) to find the horizontal and vertical shifts.
Question 4 describes a cube root function reflected over the x-axis, stretched vertically by a factor of 3, and shifted down by 2. All these transformations are incorporated into the function as: multiply by -3 and subtract 2.
Understanding these transformations helps in graphing and interpreting the behavior of radical functions.
