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.