Match the parabolic equations with their opening directions.

1. [tex]y = -5x^{\wedge}2 + 10x - 3[/tex]
a. Opens downwards

2. [tex]y = 3x^{\wedge}2 + 6x + 2[/tex]
b. Opens upwards

3. [tex]y = 0.5x^{\wedge}2 - x + 1[/tex]
c. Opens upwards



Answer :

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.