Problem 74 - College Algebra

Question

What is a quadratic function?

Step-by-Step Solution

Answer

A quadratic function is a function that can be described by an equation of the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a \) is not zero. The graph of a quadratic function is a parabola.

Step 1: Definition of Quadratic Function

A quadratic function is a function that can be written in the standard form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are real numbers and \( a \) is not equal to zero. The graph of a quadratic function is called a parabola.

Step 2: Features of Quadratic Functions

In a quadratic function, the highest power of the variable (usually denoted as \( x \)) is 2. The standard form of the function allows us to calculate key features: the direction of the parabola (upward if \( a > 0 \), downward if \( a < 0 \)), the vertex (the maximum or minimum point), and the function's zeroes (the x-values where \( f(x) = 0 \)).

Step 3: Examples of Quadratic Functions

Examples of quadratic functions include \( f(x) = 2x^2 + 3x + 1 \), where \( a = 2, b = 3, c = 1 \), and \( g(x) = -x^2 + 4x - 5 \), where \( a = -1, b = 4, c = -5 \). Both are valid quadratic functions, as their highest power of \( x \) is 2.