https://file.aiquickdraw.com/mathbot/file/1726023723222wmb93eq0.jpg

Let's work through these problems step-by-step.

### Problem 11: 
Parent function appears to be \( f(x) = |x| \) (absolute value function).
- **a) Parent Function**: \( f(x) = |x| \)
- **b) Translate 4 units right**: The function becomes \( f(x) = |x - 4| \)
- **c) Reflect in the x-axis and translate 5 units up**: Reflecting in the x-axis changes the function to \( f(x) = -|x| \), then translating 5 units up gives \( f(x) = -|x| + 5 \)

### Problem 12:
Parent function appears to be \( f(x) = \frac{1}{x} \) (rational function).
- **a) Parent Function**: \( f(x) = \frac{1}{x} \)
- **b) Translate 7 units down and 3 units left**: The function becomes \( f(x) = \frac{1}{x + 3} - 7 \)
- **c) Horizontally stretch by a factor of 3 and translate 2 units right**: Horizontally stretching changes the function to \( f(x) = \frac{1}{\frac{x}{3}} = \frac{3}{x} \), and then translating 2 units right gives \( f(x) = \frac{3}{x - 2} \)

### Problem 13:
Parent function appears to be \( f(x) = x \) (linear function).
- **a) Parent Function**: \( f(x) = x \)
- **b) Translate 1 unit up and vertically stretch by a factor of 4**: First stretch vertically to get \( f(x) = 4x \), then translate 1 unit up: \( f(x) = 4x + 1 \)
- **c) Vertically compress by a factor of 1/3 and translate 6 units left**: Vertically compress to get \( f(x) = \frac{1}{3}x \), then translate 6 units left: \( f(x) = \frac{1}{3}(x + 6) \)

### Problem 14:
Parent function appears to be \( f(x) = \sqrt{x} \) (square root function).
- **a) Parent Function**: \( f(x) = \sqrt{x} \)
- **b) Reflect in the y-axis and translate 4 units right**: Reflecting in the y-axis changes the function to \( f(x) = \sqrt{-x} \), then translating 4 units right gives \( f(x) = \sqrt{-(x - 4)} \)
- **c) Horizontally compress by a factor of 1/2 and translate 5 units down**: Horizontally compress to get \( f(x) = \sqrt{2x} \), then translate 5 units down: \( f(x) = \sqrt{2x} - 5 \)

### Problem 15:
You are given \( f(x) = 2(x - 5)^2 + 3 \). First, reflect across the y-axis, which gives \( f(x) = 2(-x - 5)^2 + 3 \). Then, translate 6 units right and 2 units down, resulting in:
\[
f(x) = 2(-(x - 6) - 5)^2 + 3 - 2
\]
Simplifying:
\[
f(x) = 2(-x + 1)^2 + 1
\]

### Problem 16:
The resulting function is \( f(x) = 6|x + 4| - 1 \). To find the original function, reverse the transformations:
1. Reverse the vertical stretch by a factor of 2: \( \frac{1}{2} \times 6 = 3 \), so \( f(x) = 3|x + 4| - 1 \)
2. Reverse the translation of 5 units up: \( f(x) = 3|x + 4| - 5 \)
3. Reverse the translation of 2 units left: \( f(x) = 3|x + 2| - 5 \)

The original function is \( f(x) = 3|x + 2| - 5 \).

---

Do you want further details on any of these transformations?  
Here are five related questions to practice:
1. How do you translate a function vertically by 7 units?
2. What is the impact of a reflection across the x-axis on a function?
3. What does it mean to horizontally stretch a function by a factor of 2?
4. How would you reflect a function over the y-axis?
5. What are the effects of compressing a function vertically?

**Tip**: When translating a function, make sure to adjust the function's input for horizontal changes and the output for vertical changes.

Transform the given parent functions by translating, reflecting, stretching, and compressing them as instructed.

This problem focuses on transforming functions through various operations such as translations, reflections, and stretches. Learn how to manipulate parent functions like absolute value, rational, linear, and square root functions in a step-by-step manner. Ideal for students in grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation