Problem 53 - Introductory Algebra for College Students

Question

Explain how to decide whether a parabola opens upward or downward.

Step-by-Step Solution

Answer

To determine whether a parabola opens upward or downward, look at the coefficient of the squared term in the parabola's equation. If the coefficient is positive, the parabola opens upward. If it's negative, the parabola opens downward.

Step 1: Understanding a Parabola

A parabola is a simple type of graph and its shape is known as a curve. Any point on this curve is equidistant from a fixed point (known as the focus) and a fixed line (the directrix). The general equation of parabola is \(y = ax^2 + bx + c\)

Step 2: Identifying Coefficient of the Squared Term

In the given equation of the parabola \(y = ax^2 + bx + c\), 'a' is the coefficient of the squared term. To determine whether the parabola opens up or down, we need to look at this coefficient.

Step 3: Deciding The Direction of Opening

If the coefficient 'a' of the squared term is positive i.e. \(a > 0\), then parabola opens upwards. On the other hand, if 'a' is negative i.e. \(a < 0\), then the parabola opens downward.