f(x)=-2|3x|+1, graph the transformations of this equation and explain them

Let's break down the function \( f(x) = -2|3x| + 1 \) step by step to identify and explain the transformations involved in the graph:

### 1. **Start with the basic function**: 
The core function is \( |x| \), which is the absolute value function. Its graph is a "V" shape with a vertex at \( (0, 0) \), opening upwards.

### 2. **Horizontal Stretch/Compression**:
The term inside the absolute value is \( |3x| \). This represents a **horizontal compression** by a factor of \( \frac{1}{3} \). 

- Normally, the graph of \( |x| \) stretches equally along the x-axis, but multiplying by 3 inside the absolute value compresses it horizontally by a factor of 3.
- Points on the graph that were originally at \( (1,1) \) and \( (-1,1) \) on \( |x| \) will now be at \( \left(\frac{1}{3},1\right) \) and \( \left(-\frac{1}{3},1\right) \).

### 3. **Vertical Stretch and Reflection**:
The coefficient \( -2 \) outside of the absolute value function has two effects:

- **Vertical stretch by a factor of 2**: Multiplying the output by 2 makes the "V" steeper. The slopes of the arms change from \( \pm 1 \) to \( \pm 2 \).
- **Reflection over the x-axis**: The negative sign flips the graph, so the "V" now opens downwards instead of upwards.

### 4. **Vertical Translation**:
The "+1" at the end of the function translates the graph **upward by 1 unit**. This means that the vertex of the graph, which was originally at \( (0, 0) \), is now moved to \( (0, 1) \).

### Summary of Transformations:
1. **Horizontal compression** by a factor of 3.
2. **Vertical stretch** by a factor of 2.
3. **Reflection** over the x-axis (opening downwards).
4. **Vertical shift** up by 1 unit.

### Graphing these transformations:
1. Start with the basic \( |x| \) graph.
2. Apply the horizontal compression: compress it by a factor of 3 along the x-axis.
3. Apply the vertical stretch and reflection: stretch it vertically and flip it to open downwards.
4. Finally, shift the entire graph up by 1 unit.

Would you like me to graph this for you or go into more details about any of the transformations?

### 5 Related Questions:
1. How does the graph change if we replace \( -2 \) with \( -1 \)?
2. What happens if the vertical shift changes from +1 to -2?
3. How would the transformations differ if we had \( |x-2| \) inside the absolute value?
4. What is the effect of multiplying the absolute value by a fraction instead of 2?
5. How can we algebraically verify the vertex and slope changes of the graph?

### Tip:
When interpreting transformations, always look for changes in signs or coefficients inside and outside the function for horizontal and vertical shifts or reflections.

Understand the graphing transformations of the equation f(x) = -2|3x| + 1. This problem covers key concepts like horizontal compression, vertical stretch, reflection over the x-axis, and vertical translation. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation