Let's go through each equation to identify which one reveals its extreme value without any alterations and determine the extreme value and its type.
### Equations Analysis:
1. Equation A: [tex]\( y = 3x^2 + 6x + 18 \)[/tex]
- This is a quadratic equation in standard form. To find its extreme value efficiently, you'd need to convert it into vertex form by completing the square, which requires some additional steps.
2. Equation B: [tex]\( y = -3(x-2)^2 + 5 \)[/tex]
- This quadratic equation is already in vertex form, [tex]\( y = a(x-h)^2 + k \)[/tex], where the vertex is [tex]\( (h, k) \)[/tex]. Here, it's clear that the vertex is at [tex]\( (2, 5) \)[/tex].
- The coefficient of the [tex]\( (x-2)^2 \)[/tex] term is negative (i.e., -3), indicating that the parabola opens downwards, leading to a maximum value at the vertex.
3. Equation C: [tex]\( y = 3(x-1)(x+3) \)[/tex]
- This is a quadratic equation in factored form. To find its extreme value, you would need to either expand the equation to standard form and then complete the square or use the vertex formula, which involves additional steps.
4. Equation D: [tex]\( y = -3x^2 + 12x \)[/tex]
- Similar to Equation A, this is a quadratic in standard form. To find the extreme value, you'd need to complete the square.
### Filling in the Statements:
Equation B reveals its extreme value without needing to be altered. The extreme value of this equation has a maximum (since it opens downwards) at the point (2, 5).
Therefore, the correct answer to complete the statements would be:
Equation B reveals its extreme value without needing to be altered. The extreme value of this equation has a maximum at the point (2, 5).
