1. Revealing Price for Zero Profit: Factored form P(x) = -2(x - 6)² + 18.
2. Price for Zero Profit: $6
3. Revealing Profit when Price is Zero: Standard quadratic form P(x) = -2(x - 3)(x - 9)
4. Profit when Price is Zero: P(0) = $54,000
5. Revealing Price for Highest Profit: Vertex form P(x) = -2(x - 6)² + 18
6. Price for Highest Profit: x = 6
1. The equivalent expression for P(x) that reveals the price which gives a profit of zero without changing the form of the expression is P(x) = -2(x - 6)² + 18. This is because the expression is in vertex form, which allows us to easily identify the x-coordinate of the vertex, which represents the price that gives a profit of zero.
2. To find the price which gives a profit of zero, we need to find the x-coordinate of the vertex of the quadratic function. In this case, the vertex is (6, 18), so the price which gives a profit of zero is $6.
3. The equivalent expression for P(x) that reveals the profit when the price is zero without changing the form of the expression is P(x) = -2(x - 3)(x - 9). This is because when the price is zero, the profit is equal to the y-intercept of the quadratic function.
4. To find the profit when the price is zero, we need to evaluate the quadratic function at x = 0. Using the equivalent expression P(x) = -2(x - 3)(x - 9), we get:
P(0) = -2(0 - 3)(0 - 9) = 54
Therefore, the profit when the price is zero is $54,000.
5. The equivalent expression for P(x) that reveals the price which produces the highest possible profit without changing the form of the expression is P(x) = -2(x - 6)² + 18. This is because the expression is in vertex form, which allows us to easily identify the x-coordinate of the vertex, which represents the price that produces the highest possible profit.
6. To find the price which gives the highest possible profit, we need to find the x-coordinate of the vertex of the quadratic function. In this case, the vertex is (6, 18), so the price which gives the highest possible profit is $6.