To determine the direction in which each parabola opens, we need to examine the coefficient of the [tex]\(x^2\)[/tex] term (also called the quadratic coefficient) in each quadratic equation. The sign of this coefficient will tell us the direction:
1. If the coefficient of the [tex]\(x^2\)[/tex] term is positive, the parabola opens upwards.
2. If the coefficient of the [tex]\(x^2\)[/tex] term is negative, the parabola opens downwards.
Let's examine each equation step by step:
1. [tex]\(y = -5x^2 + 10x - 3\)[/tex]
- The coefficient of [tex]\(x^2\)[/tex] is [tex]\(-5\)[/tex].
- Since [tex]\(-5\)[/tex] is less than 0, the parabola opens downwards.
- Therefore, this equation matches with "a. Opens downwards".
2. [tex]\(y = 3x^2 + 6x + 2\)[/tex]
- The coefficient of [tex]\(x^2\)[/tex] is [tex]\(3\)[/tex].
- Since [tex]\(3\)[/tex] is greater than 0, the parabola opens upwards.
- Therefore, this equation matches with "b. Opens upwards".
3. [tex]\(y = 0.5x^2 - x + 1\)[/tex]
- The coefficient of [tex]\(x^2\)[/tex] is [tex]\(0.5\)[/tex].
- Since [tex]\(0.5\)[/tex] is greater than 0, the parabola opens upwards.
- Therefore, this equation matches with "c. Opens upwards".
So, the matching directions for the given parabolic equations are:
1. [tex]\(y = -5x^2 + 10x - 3\)[/tex]
- Opens downwards.
2. [tex]\(y = 3x^2 + 6x + 2\)[/tex]
- Opens upwards.
3. [tex]\(y = 0.5x^2 - x + 1\)[/tex]
- Opens upwards.
