To determine the direction in which the parabola opens given the quadratic equation [tex]\( y = -x^2 + 2x - 7 \)[/tex], we need to look at the coefficient of the [tex]\( x^2 \)[/tex] term in the standard form of a quadratic equation.
The standard form of a quadratic equation is:
[tex]\[
y = ax^2 + bx + c
\][/tex]
Here, [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants.
For the given equation:
[tex]\[
y = -x^2 + 2x - 7
\][/tex]
- The coefficient of [tex]\( x^2 \)[/tex] is [tex]\( a = -1 \)[/tex].
- The coefficient of [tex]\( x \)[/tex] is [tex]\( b = 2 \)[/tex].
- The constant term is [tex]\( c = -7 \)[/tex].
The direction in which a parabola opens is determined by the sign of the coefficient [tex]\( a \)[/tex]:
- If [tex]\( a > 0 \)[/tex], the parabola opens upward.
- If [tex]\( a < 0 \)[/tex], the parabola opens downward.
In this particular case, the coefficient [tex]\( a \)[/tex] is [tex]\( -1 \)[/tex], which is less than zero ([tex]\( a < 0 \)[/tex]). Thus,
[tex]\[
\boxed{\text{down}}
\][/tex]
This tells us that the parabola opens down.
