Question

y = f(x) and it’s translated form y = f(x) + k has undergone a vertical shift what would k be

Answer 1

Points on original (-1,-1),(0,0),and (1,1)solid line on the vertical shift which is a dotted line has points (-1,-3),(0,-2) and (1,-1)

Answer 2

In the context of the function \(y = f(x)\) being translated to \(y = f(x) + k\), the constant \(k\) represents the vertical shift of the function.

- If \(k > 0\), the graph of the function \(f(x)\) shifts upward by \(k\) units.
- If \(k < 0\), the graph shifts downward by \(|k|\) units.
- If \(k = 0\), there is no vertical shift.

To summarize, \(k\) is the value that determines how much the function is shifted vertically from its original position.

Answer 3

To analyze the vertical shift from the original points to the new points, we can compare the corresponding \(y\)-coordinates of the points.

### Original Points:
1. \((-1, -1)\)
2. \((0, 0)\)
3. \((1, 1)\)

### Shifted Points:
1. \((-1, -3)\)
2. \((0, -2)\)
3. \((1, -1)\)

### Finding the Shift:
To find the vertical shift \(k\), we can look at the changes in the \(y\)-coordinates for the corresponding \(x\)-values.

1. For \((-1, -1)\) to \((-1, -3)\):
- Shift: \(-3 - (-1) = -3 + 1 = -2\)

2. For \((0, 0)\) to \((0, -2)\):
- Shift: \(-2 - 0 = -2\)

3. For \((1, 1)\) to \((1, -1)\):
- Shift: \(-1 - 1 = -2\)

### Conclusion:
The vertical shift that takes the original line described by the points to the new line is consistently \(k = -2\). This means the function has been shifted downward by 2 units.

Thus, the relationship can be expressed as:
\[
y = f(x) - 2
\] where \(k = -2\).