To determine the direction that the parabola with the equation [tex]\( y = -x^2 + 6x - 13 \)[/tex] opens, we need to examine the coefficient of the [tex]\( x^2 \)[/tex] term.
The general form of a quadratic equation is given by:
[tex]\[
y = ax^2 + bx + c
\][/tex]
where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants.
In our equation:
[tex]\[
y = -x^2 + 6x - 13
\][/tex]
we can clearly see that:
- The coefficient [tex]\( a \)[/tex] for [tex]\( x^2 \)[/tex] is [tex]\(-1\)[/tex].
- The coefficient [tex]\( b \)[/tex] for [tex]\( x \)[/tex] is [tex]\(6\)[/tex].
- The constant term [tex]\( c \)[/tex] is [tex]\(-13\)[/tex].
The direction in which the parabola opens is determined by the sign of the coefficient [tex]\( a \)[/tex]:
1. If [tex]\( a \)[/tex] is positive ([tex]\( a > 0 \)[/tex]), the parabola opens upwards.
2. If [tex]\( a \)[/tex] is negative ([tex]\( a < 0 \)[/tex]), the parabola opens downwards.
In our specific equation, the coefficient [tex]\( a \)[/tex] is [tex]\(-1\)[/tex], which is negative. Therefore, the parabola opens downwards.
So, the correct answer is:
down
