1. **State the problem:** We are given the profit function for a company producing phones: $$P(x) = -1.5x^2 + 10.5x - 4$$ where $x$ is the number of phones produced in tens of thousands, and $P(x)$ is the profit in millions of euro. We need to find $P(0)$ and interpret it, complete a table of values, and understand the graph. 2. **Find $P(0)$:** Substitute $x=0$ into the profit function: $$P(0) = -1.5(0)^2 + 10.5(0) - 4 = -4$$ 3. **Interpretation:** $P(0) = -4$ means that if the company produces zero phones, it will have a loss of 4 million euro in the first year. This could represent fixed costs or losses without production. 4. **Complete the table:** Calculate $P(x)$ for $x=1,2,3,4,5,6,7$ using the formula: - $P(1) = -1.5(1)^2 + 10.5(1) - 4 = -1.5 + 10.5 - 4 = 5$ - $P(2) = -1.5(4) + 10.5(2) - 4 = -6 + 21 - 4 = 11$ - $P(3) = -1.5(9) + 10.5(3) - 4 = -13.5 + 31.5 - 4 = 14$ - $P(4) = -1.5(16) + 10.5(4) - 4 = -24 + 42 - 4 = 14$ - $P(5) = -1.5(25) + 10.5(5) - 4 = -37.5 + 52.5 - 4 = 11$ - $P(6) = -1.5(36) + 10.5(6) - 4 = -54 + 63 - 4 = 5$ - $P(7) = -1.5(49) + 10.5(7) - 4 = -73.5 + 73.5 - 4 = -4$ 5. **Summary table:** | $x$ (tens of thousands) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |-------------------------|---|---|---|---|---|---|---|---| | $P(x)$ (millions euro) | -4| 5 |11 |14 |14 |11 | 5 | -4| 6. **Graph description:** The graph of $P(x)$ is a downward-opening parabola starting at $P(0)=-4$, rising to a maximum profit of 14 million euro at $x=3$ and $x=4$, then decreasing back to $-4$ at $x=7$. The parabola is symmetric about the vertex between $x=3$ and $x=4$. This completes the solution for part (a).