To determine how the quadratic function [tex]\( y = -3x^2 + 7x - 2 \)[/tex] behaves, we need to analyze its components.
### 1. Identifying the Direction of Parabola Opening
A quadratic function is generally represented as [tex]\( y = ax^2 + bx + c \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants.
- If [tex]\( a > 0 \)[/tex], the parabola opens upwards.
- If [tex]\( a < 0 \)[/tex], the parabola opens downwards.
In the given function [tex]\( y = -3x^2 + 7x - 2 \)[/tex]:
- The coefficient of [tex]\( x^2 \)[/tex] is [tex]\( -3 \)[/tex].
- Since [tex]\( -3 \)[/tex] is less than zero ( [tex]\( -3 < 0 \)[/tex] ), the parabola opens downwards.
### 2. Identifying the Vertex Type
The vertex of a parabola represents the maximum or minimum point of the function.
- If the parabola opens upwards ( [tex]\( a > 0 \)[/tex] ), the vertex is a minimum point.
- If the parabola opens downwards ( [tex]\( a < 0 \)[/tex] ), the vertex is a maximum point.
In the given function:
- Since the parabola opens downwards ( [tex]\( -3 < 0 \)[/tex] ), the vertex is a maximum point.
### Conclusion
Given the function [tex]\( y = -3x^2 + 7x - 2 \)[/tex]:
- The parabola opens down due to the negative coefficient of [tex]\( x^2 \)[/tex].
- The vertex is a maximum point since the parabola opens down.
Therefore, the answer is:
opens down with a maximum
