To determine the correct statements about the given equation of the parabola [tex]\( f(x) = -2x^2 + 12x + 21 \)[/tex], we break down the problem into a few steps.
Step 1: Determine if the parabola opens up or down
The general form of a parabolic equation is [tex]\( f(x) = ax^2 + bx + c \)[/tex], where the sign of [tex]\( a \)[/tex] determines the direction in which the parabola opens:
- If [tex]\( a > 0 \)[/tex], the parabola opens up.
- If [tex]\( a < 0 \)[/tex], the parabola opens down.
In the given equation, [tex]\( a = -2 \)[/tex]. Since [tex]\( a \)[/tex] is negative, the parabola opens down.
Step 2: Calculate the vertex of the parabola
The vertex of a parabola given by the equation [tex]\( f(x) = ax^2 + bx + c \)[/tex] can be found using the vertex formula:
- The x-coordinate of the vertex is given by [tex]\( x = -\frac{b}{2a} \)[/tex].
- The y-coordinate of the vertex is found by substituting this x-value back into the equation.
Given the coefficients:
- [tex]\( a = -2 \)[/tex]
- [tex]\( b = 12 \)[/tex]
- [tex]\( c = 21 \)[/tex]
First, find the x-coordinate of the vertex:
[tex]\[ x = -\frac{b}{2a} = -\frac{12}{2(-2)} = -\frac{12}{-4} = 3 \][/tex]
Next, substitute [tex]\( x = 3 \)[/tex] back into the equation to find the y-coordinate:
[tex]\[ f(3) = -2(3)^2 + 12(3) + 21 \][/tex]
[tex]\[ f(3) = -2(9) + 36 + 21 \][/tex]
[tex]\[ f(3) = -18 + 36 + 21 \][/tex]
[tex]\[ f(3) = 39 \][/tex]
So, the vertex is [tex]\( (3, 39) \)[/tex].
Conclusion
Given the analysis above:
1. The parabola opens down.
2. The vertex is not [tex]\( (-3,-3) \)[/tex].
3. The vertex is [tex]\( (3,39) \)[/tex], not [tex]\( (3,3) \)[/tex].
Thus, the correct statements are:
- The parabola opens down.
