Problem 90 - Precalculus : Building Concepts and Connections

Question

This set of exercises will draw on the ideas presented in this section and your general math background. Why must we have \(a \neq 0\) in the definition of a quadratic function?

Step-by-Step Solution

Answer

We must have \(a \neq 0\) in the definition of a quadratic function to ensure that the function is not linear and that it represents a parabolic shape. If \(a = 0\), the function would become a linear function, not a quadratic function.

Step 1: Understand the Structure of a Quadratic Function

A quadratic function is defined by the equation \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a \neq 0\). If \(a = 0\), then the equation simplifies to \(f(x) = bx + c\), which is a linear equation, not a quadratic.

Step 2: Understand the Importance of Term \(ax^2\)

The term \(ax^2\) determines the curvature of the graph of the function. Its coefficient \(a\) influences whether the graph opens upward \(a > 0\) or downward \(a < 0\). If \(a = 0\), the graph would be a straight line, which is not characteristic of a quadratic function.

Step 3: Conclusion

We must have \(a \neq 0\) in the definition of a quadratic function in order for the equation to represent a parabola. If \(a = 0\), the equation would be for a linear function, not a quadratic function. Therefore, \(a \neq 0\) is a necessary condition for a quadratic function.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that is defined by its degree, which is the highest power of the variable involved. In the case of quadratic equations, the degree is 2.
The general form of a quadratic equation is expressed as:

• \( ax^2 + bx + c = 0 \)

Here, \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. The key feature that distinguishes it as quadratic is the \(ax^2\) term, where \(a eq 0\).

• If \(a = 0\), the equation reduces to a linear equation, \(bx + c = 0\).

This reduction indicates a straight line graph rather than a curve. Hence, for an equation to qualify as quadratic, the term with \(x^2\) must be present and non-zero. This term is responsible for the characteristic parabolic graph.

Parabola

In the study of quadratic functions, the graph of any quadratic equation results in a curve known as a parabola. The parabolic shape is a key characteristic of quadratic equations.
Parabolas can open upward or downward based on the sign of the leading coefficient, \(a\), in the quadratic function:

• If \(a > 0\), the parabola opens upward, resembling a U-shape.
• If \(a < 0\), it opens downward, forming an upside-down U.

The vertex of the parabola is the point where it changes direction, and it is either the highest or lowest point on the graph, depending on the direction the parabola opens. Additionally, the parabola is symmetric around a vertical line that passes through its vertex, known as the axis of symmetry. This symmetry and direction are direct results of the presence of the squared term \(ax^2\) in the quadratic equation.

Linear Function

A linear function is the simplest form of a polynomial, characterized by its degree of 1. It takes the form:

• \(f(x) = bx + c\)

The essence of a linear function is its constant rate of change, resulting in a straight line when graphed. This is in contrast to the curve of a quadratic function.
The absence of the \(x^2\) term is crucial; when you have a quadratic equation \(ax^2 + bx + c\) but \(a = 0\), it simplifies to a linear equation. Consequently, the function no longer exhibits the properties of a parabola.
Linear functions have constant slopes that determine the steepness and direction of the line:

• The slope is \(b\), which tells you how much \(y\) increases or decreases as \(x\) increases.

Unlike parabolas, linear functions don’t have a vertex or axis of symmetry. Thus, they are uniquely straightforward in behavior and graph representation, distinguishing them clearly from quadratic functions.