To determine which statements are true for the given equation of the parabola [tex]\( f(x) = -2x^2 + 12x + 21 \)[/tex], we need to follow these steps:
1. Identifying the Coefficients:
- The equation is in the standard form [tex]\( ax^2 + bx + c \)[/tex].
- Here, [tex]\( a = -2 \)[/tex], [tex]\( b = 12 \)[/tex], and [tex]\( c = 21 \)[/tex].
2. Finding the Vertex:
- The x-coordinate of the vertex of a parabola [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formula [tex]\( x = -\frac{b}{2a} \)[/tex].
- Substitute [tex]\( a = -2 \)[/tex] and [tex]\( b = 12 \)[/tex] into the formula:
[tex]\[
x = -\frac{12}{2 \times -2} = -\frac{12}{-4} = 3
\][/tex]
- Next, calculate the y-coordinate of the vertex by substituting [tex]\( x = 3 \)[/tex] back into the original equation:
[tex]\[
y = -2(3)^2 + 12(3) + 21 = -2(9) + 36 + 21 = -18 + 36 + 21 = 39
\][/tex]
- So, the vertex of the parabola is [tex]\( (3, 39) \)[/tex].
3. Direction of the Parabola:
- The direction in which the parabola opens is determined by the sign of the coefficient [tex]\( a \)[/tex].
- If [tex]\( a < 0 \)[/tex], the parabola opens downwards.
- If [tex]\( a > 0 \)[/tex], the parabola opens upwards.
- In this case, since [tex]\( a = -2 \)[/tex] (which is less than 0), the parabola opens downwards.
4. Verifying the Statements:
- "The vertex is [tex]\( (3,3) \)[/tex]": False. The vertex is [tex]\( (3,39) \)[/tex], not [tex]\( (3,3) \)[/tex].
- "The parabola opens down": True. Because [tex]\( a = -2 \)[/tex] which is less than 0.
- "The vertex is [tex]\( (-3,-3) \)[/tex]": False. The vertex is [tex]\( (3,39) \)[/tex], not [tex]\( (-3,-3) \)[/tex].
- "The parabola opens up": False. The parabola opens downwards.
In conclusion, the correct statements are:
- The parabola opens down.
