Problem 165 - Elementary Algebra

Question

In the following exercises, determine if the parabola opens up or down. $$ y=-2 x^{2}-6 x-7 $$

Step-by-Step Solution

Answer

The parabola opens downward.

Step 1: Identify the Coefficient of the Quadratic Term

Look at the coefficient of the term with the highest degree, which is the quadratic term (the term with the variable raised to the power of 2). In the given equation, the quadratic term is y = -2 x^2 - 6 x - 7, and the coefficient of this term is -2.

Step 2: Determine the Direction Based on the Coefficient

The direction in which the parabola opens depends on the sign of the coefficient of the quadratic term. If the coefficient is positive, the parabola opens upward. If the coefficient is negative, the parabola opens downward. In this case, the coefficient is -2, which is negative.

Step 3: Conclusion

Since the coefficient of the quadratic term (-2) is negative, the parabola opens downward.

Key Concepts

Parabola Direction

The direction in which a parabola opens is one of the fundamental aspects you need to understand when dealing with quadratic equations. In a general quadratic equation of the form y = ax^2 + bx + c .. the term with the highest degree ( ax^2 ) plays a crucial role. The sign of the coefficient (a) tells us whether the parabola opens upwards or downwards.

If the coefficient (a) is positive, the parabola will open upward.

If the coefficient (a) is negative, the parabola will open downward.

Understanding this concept is important because it helps us quickly visualize the graph of the quadratic equation without plotting all the points.

Quadratic Term

The quadratic term is the part of the equation that contains the variable raised to the second power (x^2). In the standard form of a quadratic equation
y = ax^2 + bx + c

the term 'ax^2' is known as the quadratic term. This term is key in determining the shape and orientation of the parabola. The quadratic term influences the steepness and width of the parabola. A larger absolute value of a means a narrower parabola, while a smaller absolute value of a results in a wider parabola. For instance, in our exercise:
y = -2x^2 - 6x - 7

The quadratic term is -2x^2, and this term is what dominates the parabola's overall shape.

Coefficient Sign

The sign of the coefficient of the quadratic term is crucial for determining the parabola's orientation. In the quadratic equation y = ax^2 + bx + c

The coefficient 'a' is the focal point. Here are key points to remember:

If 'a' is positive ( a > 0
), the parabola opens upward, creating a U-shape.

If 'a' is negative (a < 0
), the parabola opens downward, forming an inverted U-shape.

This simple rule helps you quickly identify the direction of the parabola upon examining the coefficient. In our example equation

y = -2x^2 - 6x - 7

The coefficient (a) is -2, which is negative. Hence, the parabola opens downward.