Let's analyze the equation [tex]\( y = -x^2 \)[/tex] to determine which description best matches its graph.
1. Identify the type of the equation:
The given equation [tex]\( y = -x^2 \)[/tex] is a quadratic equation. Quadratic equations generally take the form [tex]\( y = ax^2 + bx + c \)[/tex].
2. Determine the key characteristics:
In this equation [tex]\( y = -x^2 \)[/tex], notice the coefficient of [tex]\( x^2 \)[/tex] is negative (specifically, -1). This is the leading coefficient and it plays a crucial role in determining the shape of the parabola.
3. Understand the effect of the negative coefficient:
For a quadratic equation of the form [tex]\( y = ax^2 \)[/tex]:
- If [tex]\( a > 0 \)[/tex] (positive), the parabola opens upwards.
- If [tex]\( a < 0 \)[/tex] (negative), the parabola opens downwards.
Here, since the coefficient [tex]\( a \)[/tex] is -1 (negative), the parabola will open downwards.
4. Analyze the shape:
A parabolic equation [tex]\( y = -x^2 \)[/tex] describes a U-shaped curve. The negative coefficient indicates this U-shape will open downwards.
Considering all this, the graph of [tex]\( y = -x^2 \)[/tex] is best described by:
c. a U-shaped curve opening downwards.
