By first completing the table of values, decide which of A-C is the graph of y = 4x...

1. Complete the table of values: We are given the function $y = 4x - x^2 - 1$ and a table with $x$ values 1, 2, and 3. We need to calculate the corresponding $y$ values.
- For $x=1$: $y = 4(1) - (1)^2 - 1 = 4 - 1 - 1 = 2$.
- For $x=2$: $y = 4(2) - (2)^2 - 1 = 8 - 4 - 1 = 3$.
- For $x=3$: $y = 4(3) - (3)^2 - 1 = 12 - 9 - 1 = 2$.
The completed table is:
| x | 1 | 2 | 3 |
|---|---|---|---|
| y | 2 | 3 | 2 |

2. Identify the graph: Now we compare these points $(1, 2)$, $(2, 3)$, and $(3, 2)$ with the given graphs A, B, and C.
- Graph A passes through $(0, -1)$, $(1, 2.5)$, $(2, 3)$, $(3, 2.5)$.
- Graph B passes through $(0, -1)$, $(1, 3)$, $(2, 4)$, $(3, 3)$.
- Graph C passes through $(0, -1)$, $(1, 2)$, $(2, 3)$, $(3, 2)$.
Graph C is the only one that passes through all the calculated points. Therefore, graph C represents the function $y = 4x - x^2 - 1$.

3. Estimate the solutions for the equation: We need to estimate the solutions to $4x - x^2 - 1 = -0.2$. This is equivalent to finding the $x$-values where the graph of $y = 4x - x^2 - 1$ intersects the horizontal line $y = -0.2$. Looking at graph C, we can see that the line $y = -0.2$ (which is slightly above $y=-0.5$) intersects the graph at two points.
- The first intersection point is to the left of the y-axis, approximately at $x = -0.2$.
- The second intersection point is to the right of the y-axis, approximately at $x = 4.2$.
Therefore, the estimated solutions are $x \approx -0.2$ and $x \approx 4.2$.