Let's analyze the given quadratic equation of the parabola:
[tex]\[ f(x) = -2x^2 + 12x + 21 \][/tex]
From this equation, we can determine several properties of the parabola.
Vertex of the Parabola:
To find the vertex of the parabola given by the equation ax^2 + bx + c, we use the formula for the x-coordinate of the vertex [tex]\( h \)[/tex]:
[tex]\[ h = -\frac{b}{2a} \][/tex]
In this equation,
[tex]\[ a = -2, \, b = 12, \, \text{and} \, c = 21 \][/tex]
Now, by applying the formula for [tex]\( h \)[/tex]:
[tex]\[ h = -\frac{12}{2(-2)} = -\frac{12}{-4} = 3 \][/tex]
Next, we find the y-coordinate [tex]\( k \)[/tex] of the vertex by substituting [tex]\( h \)[/tex] back into the equation:
[tex]\[ k = a(3)^2 + b(3) + c \][/tex]
[tex]\[ k = -2(3)^2 + 12(3) + 21 \][/tex]
[tex]\[ k = -2(9) + 36 + 21 \][/tex]
[tex]\[ k = -18 + 36 + 21 \][/tex]
[tex]\[ k = 39 \][/tex]
Thus, the vertex of the parabola is [tex]\( (3, 39) \)[/tex].
Direction in which the Parabola Opens:
The coefficient of the [tex]\( x^2 \)[/tex] term, [tex]\( a \)[/tex], determines whether the parabola opens up or down:
- If [tex]\( a \)[/tex] is positive, the parabola opens up.
- If [tex]\( a \)[/tex] is negative, the parabola opens down.
In this case, [tex]\( a = -2 \)[/tex], which is negative. Therefore, the parabola opens down.
Verifying the Statements:
Let's evaluate each statement given in the question:
1. The vertex is [tex]\((3,3)\)[/tex]: This statement is false, as we have calculated that the vertex is [tex]\((3, 39)\)[/tex].
2. The parabola opens down: This statement is true because [tex]\( a = -2 \)[/tex] is negative.
3. The parabola opens up: This statement is false because [tex]\( a = -2 \)[/tex] is negative.
4. The vertex is [tex]\((-3,-3)\)[/tex]: This statement is false, as we have calculated the vertex to be [tex]\((3, 39)\)[/tex].
Conclusion:
The correct statement from the options given is:
- The parabola opens down.
