The graphs of polynomial functions f (x) and g(x) = f (x) + k are given.

A graph of a polynomial labeled as f of x decreases from the second quadrant passing through negative 3,4 and negative 2,0 to approximately negative 0.8, negative 1.4 and increases through 0,negative 1 and 1,0 to approximately 2,0.7 and decreases through 3,0 and 4, negative 3 and continues to decrease without bound.   A graph of a polynomial labeled as g of x decreases from the second quadrant passing through negative 3,2 and negative 2,negative 2 to approximately negative 0.8, negative 3,4 and increases through 0, negative 3 and 1, negative 2 to approximately 2, negative 1.3 and decreases through 3, negative 2 and approximately 4,negative 5 and continues to decrease without bound.

Determine the additive transformation.

 Horizontal translation left two units
 Horizontal translation right two units
 Vertical translation up two units
 Vertical translation down two units

To determine the additive transformation between the functions \( f(x) \) and \( g(x) = f(x) + k \), we can compare the vertical positions of corresponding points on the graphs of \( f(x) \) and \( g(x) \).

### Key Observations:
- Both graphs have similar shapes, but the graph of \( g(x) \) appears to be shifted vertically compared to \( f(x) \).
- The points on \( g(x) \) are consistently 2 units lower than the corresponding points on \( f(x) \).

### Analysis:
- At \( x = -3 \), \( f(x) \) passes through \( (-3, 4) \) and \( g(x) \) passes through \( (-3, 2) \).
- At \( x = -2 \), \( f(x) \) passes through \( (-2, 0) \) and \( g(x) \) passes through \( (-2, -2) \).
- At \( x = 0 \), \( f(x) \) passes through \( (0, -1) \) and \( g(x) \) passes through \( (0, -3) \).

In each case, \( g(x) \) is 2 units lower than \( f(x) \). 

### Conclusion:
The graph of \( g(x) \) is obtained by vertically translating the graph of \( f(x) \) down by 2 units. Therefore, the correct transformation is:

**Vertical translation down two units.**

Would you like further details or have any questions?

### Follow-up Questions:
1. How would the transformation change if \( g(x) = f(x) + 3 \)?
2. What would be the effect of a horizontal translation of \( f(x) \)?
3. How can you determine if a transformation is horizontal or vertical?
4. How does adding a constant \( k \) to a function affect its graph?
5. What is the impact on the graph if \( g(x) = f(x) - k \)?

**Tip:** Always observe the corresponding points on the graphs when comparing transformations.

The graphs of polynomial functions f(x) and g(x) = f(x) + k are given. Determine the additive transformation.

Explore the transformation of polynomial graphs with a detailed example showing the effect of a vertical translation. Learn how adding a constant to a function affects its graph, with a focus on understanding vertical shifts and transformations. Suitable for high school students in Grades 9-12.

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