To find the transformation between the original function $f(x)$ and the new function, we need to analyze how the points have changed.

Original Points of $f(x)$:

• $(-4, 4)$
• $(-2, 0)$
• $(0, -4)$
• $(4, 4)$

New Points:

• $(0, -2)$
• $(2, -2)$
• $(4, 2)$
• $(8, 2)$

Step 1: Analyze the transformation

• First, observe the $x$-coordinates.

  • In the original points, $x = -4$ is mapped to $x = 0$, $x = -2$ is mapped to $x = 2$, $x = 0$ to $x = 4$, and $x = 4$ to $x = 8$.

  • This suggests a horizontal shift and scaling of the $x$-coordinates.

    By observing these shifts, we can determine that the $x$-coordinates have been transformed by the rule: $x_{\text{new}} = 2x + 8$ This involves a horizontal scaling by 2 and a rightward shift by 8 units.

• Now, let's analyze the $y$-coordinates.

  • In the original function, the $y$-coordinates are $4, 0, -4, 4$.

  • In the transformed function, the $y$-coordinates are $-2, -2, 2, 2$.

    The pattern shows that the $y$-coordinates have been scaled and shifted. It appears that: $y_{\text{new}} = \frac{y}{2} + 2$ This involves a vertical compression by a factor of 2 and a vertical shift upwards by 2 units.

Step 2: Write the transformation equation

From the above analysis, the transformation of the function $f(x)$ to the new function involves:

• A horizontal scaling by 2 and a rightward shift by 8 units.
• A vertical compression by a factor of 2 and a shift upward by 2 units.

Thus, the transformed function $g(x)$ can be expressed as: $g(x) = \frac{1}{2}f\left(\frac{x-8}{2}\right) + 2$

Summary:

The equation for the transformation is: $g(x) = \frac{1}{2}f\left(\frac{x-8}{2}\right) + 2$

Would you like more details, or do you have any questions?

Here are 5 related questions to explore further:

1. How would the transformation change if the vertical shift was downward instead of upward?
2. What would the transformation look like if there was no horizontal scaling?
3. How would a reflection across the y-axis affect the transformation equation?
4. How do you interpret the effect of a transformation on the graph of a function visually?
5. What happens if the original function $f(x)$ is a specific known function like a parabola?

Tip: To identify transformations, always analyze the changes in both the x- and y-coordinates separately—this helps in detecting scaling, shifting, and reflecting patterns.