https://file.aiquickdraw.com/mathbot/file/1729302642106mb9cxadh.jpg

The image you've uploaded shows a problem related to the transformation of linear functions. The goal is to identify the transformations applied to the basic linear function \( f(x) = x \) to produce the transformed function \( g(x) = -0.5x + 1 \).

Let's break down the transformation step-by-step:

1. **Reflection**: The negative sign in front of the coefficient of \(x\) in \( g(x) = -0.5x + 1 \) suggests a reflection over the x-axis. This is because multiplying by a negative number reflects the graph vertically.

2. **Dilation** (Vertical Scaling): The coefficient of \(x\) is \( -0.5 \), which means that the graph of the basic function \( f(x) = x \) is vertically compressed by a factor of 0.5. This is a vertical scaling transformation that shrinks the slope of the line.

3. **Vertical Translation**: The constant term \( +1 \) indicates a vertical shift. This translates the entire graph of the function upwards by 1 unit.

### Summary of Transformations:
- **Reflection** over the x-axis.
- **Vertical Dilation** by a factor of 0.5.
- **Vertical Translation** up by 1 unit.

Would you like further clarification on any of these transformations?

Here are five related questions to deepen your understanding:
1. How would the transformation change if the coefficient of \(x\) was positive?
2. What happens to the graph of the function if the vertical translation was negative?
3. How does a horizontal translation differ from a vertical translation in linear functions?
4. How does the slope of a linear function affect the steepness of the graph?
5. How would you apply multiple transformations in a different order, such as reflecting after translating?

**Tip**: Remember that reflections change the direction of the graph, but dilations modify the steepness or flatness of the slope.

The equation for the transformed function is g(x) = -0.5x + 1. Indicate the transformation(s) of the basic linear function f(x) = x needed to create g(x).

Explore how to transform the basic linear function f(x) = x into the function g(x) = -0.5x + 1. Learn about the key transformations including reflection over the x-axis, vertical scaling by 0.5, and vertical translation by 1 unit. This problem covers essential concepts for 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