Question

Original function y =f(x)
Graph downward with point (0,0), (-1,-1),(1,1)(-2,-4),(2,-4)
The translated function y = f(x) + k
Downward graph with point at (0,-3), (-1,-4), (1,-4)
What is the value if k

Answer 1

To determine the value of \( k \) for the translated function \( y = f(x) + k \), we can analyze the points provided:

1. **Original function points**:
- \( (0, 0) \)
- \( (-1, -1) \)
- \( (1, 1) \)
- \( (-2, -4) \)
- \( (2, -4) \)

2. **Translated function points**:
- \( (0, -3) \)
- \( (-1, -4) \)
- \( (1, -4) \)

We are told the new function is given by \( y = f(x) + k \). This means that every y-coordinate of the original function will be increased by \( k \).

Let's consider what happens to the point \( (0, 0) \):
- In the original function: \( f(0) = 0 \)
- In the translated function: \( f(0) + k = -3 \)

Setting these equal gives us:
\[
0 + k = -3
\]
Thus:
\[
k = -3
\]

Now, let’s confirm that this value of \( k \) also aligns with the other points:
- For the point \( (-1, -1) \):
- In the translated function: \( f(-1) + k = -4 \)
- So, \(-1 + k = -4 \Rightarrow -1 - 3 = -4\), which holds true.

- For the point \( (1, 1) \):
- In the translated function: \( f(1) + k = -4 \)
- So, \(1 + k = -4 \Rightarrow 1 - 3 = -4\), which also holds true.

The value of \( k \) is confirmed to be \( -3 \), as it satisfies the translation for all provided points.

Thus, the value of \( k \) is \( \boxed{-3} \).