To determine whether the parabola defined by the quadratic equation [tex]\( y = -3x^2 + 7x - 2 \)[/tex] opens up or down, we need to analyze the coefficient of the [tex]\( x^2 \)[/tex] term.
A quadratic equation is typically written in the general form:
[tex]\[ y = ax^2 + bx + c \][/tex]
where:
- [tex]\( a \)[/tex] is the coefficient of [tex]\( x^2 \)[/tex],
- [tex]\( b \)[/tex] is the coefficient of [tex]\( x \)[/tex],
- [tex]\( c \)[/tex] is the constant term.
The direction in which the parabola opens is determined by the coefficient [tex]\( a \)[/tex]:
1. If [tex]\( a > 0 \)[/tex], the parabola opens upwards.
2. If [tex]\( a < 0 \)[/tex], the parabola opens downwards.
For the equation [tex]\( y = -3x^2 + 7x - 2 \)[/tex]:
- The coefficient [tex]\( a \)[/tex] is [tex]\(-3\)[/tex].
Since [tex]\( a = -3 \)[/tex] is less than zero, the parabola opens downwards.
Conclusion: The parabola opens down.
