The primary difference between a quadratic function and a quadratic equation lies in their structure and what they represent: a quadratic function expresses a relationship, while a quadratic equation sets that relationship equal to a specific value (often zero) to solve for the variable. According to our reference, the quadratic equation is a mathematical statement, which has an equal sign and has a value of zero, meanwhile, the value of the quadratic function can be zero and non-zero. Also, the quadratic equation has an equal sign, whereas the quadratic function does not have.
Here's a breakdown:
Quadratic Function
- Definition: A quadratic function is a polynomial function of degree two. It describes a relationship between an input (x) and an output (y), and its graph is a parabola.
- General Form: f(x) = ax2 + bx + c, where 'a', 'b', and 'c' are constants and 'a' ≠ 0. The output, f(x), can also be represented as y.
- Purpose: Describes a curve (parabola) and the relationship between x and y. You can analyze its properties like vertex, axis of symmetry, and intercepts. The function's value (f(x) or y) can be any real number.
- No Equal Sign (Initially): It expresses a relationship f(x) depends on x.
Quadratic Equation
- Definition: A quadratic equation is a statement that sets a quadratic expression (the same as the right-hand side of a quadratic function) equal to a value, typically zero.
- General Form: ax2 + bx + c = 0, where 'a', 'b', and 'c' are constants and 'a' ≠ 0.
- Purpose: To find the value(s) of 'x' that satisfy the equation; i.e., the value(s) of 'x' that make the expression equal to zero. These values are called roots or solutions.
- Has an Equal Sign: This is the key differentiator. You're solving for x where ax2 + bx + c equals zero.
Summary in a Table
| Feature | Quadratic Function | Quadratic Equation |
|---|---|---|
| Definition | A polynomial function of degree two. | A statement setting a quadratic expression equal to a value. |
| General Form | f(x) = ax2 + bx + c | ax2 + bx + c = 0 |
| Equal Sign | No (until you evaluate f(x) for a specific x) | Yes |
| Purpose | Describes a relationship; graphed as a parabola. | To find the value(s) of 'x' that satisfy the equation. |
| Solutions/Roots | Not applicable (it's a function, not an equation) | Solutions are the 'x' values that make the equation true. |
| Value of expression | Can be zero and non-zero. | Has a value of zero. |
Example
- Quadratic Function: f(x) = x2 - 4x + 3
- Corresponding Quadratic Equation: x2 - 4x + 3 = 0
The function describes a parabola. To find where the parabola intersects the x-axis, we solve the equation x2 - 4x + 3 = 0, which gives us x = 1 and x = 3. These are the roots (or solutions) of the equation.
