To solve the transformation problem depicted in the image, let's analyze the given graph and the parent function.

Step 1: Identify the Parent Function

The parent function appears to be $y = |x|$, which is the absolute value function. This is indicated by the V-shaped graph.

Step 2: Identify the Transformations

We need to determine how the parent function $y = |x|$ is transformed into the red dashed graph.

• Vertical Stretch/Compression: Compare the slope of the lines. The parent function has a slope of $1$ (or $-1$ on the left side), but the red dashed graph has a slope of $\frac{3}{2}$ (or $-\frac{3}{2}$ on the left side). This indicates a vertical stretch by a factor of $\frac{3}{2}$.

• Horizontal Translation: There is no horizontal shift, as the vertex remains at the origin $(0, 0)$.

• Vertical Translation: The red graph is shifted downward by 2 units, so a vertical translation of $-2$ is applied.

Step 3: Write the Transformation Equation

The transformed equation for the red graph, based on the transformations identified, is:

$g(x) = \frac{3}{2}f(x) - 2$

Substituting the parent function $f(x) = |x|$:

$g(x) = \frac{3}{2}|x| - 2$

This equation should transform the parent function into the red dashed graph.

Would you like more details or have any questions about the solution?

Further Questions:

1. How would the graph look if the parent function was $y = x^2$ instead of $y = |x|$?
2. What would happen if the graph were translated 3 units to the right?
3. How would the equation change if the graph were stretched horizontally instead of vertically?
4. What is the effect of changing the vertical translation from $-2$ to $+2$?
5. How do you determine the vertex of the transformed graph?

Tip:

When analyzing transformations, always start with identifying basic shifts (vertical/horizontal translations) before moving to stretches, compressions, or reflections.