๐ What is a Quadratic Function?
A quadratic function is a polynomial function of degree 2. The general form of a quadratic function is:
$f(x) = ax^2 + bx + c$,
where $a$, $b$, and $c$ are constants, and $a \neq 0$. The graph of a quadratic function is a parabola.
- ๐ Shape: The graph is a parabola, which is a U-shaped curve.
- ๐ Vertex: It has a vertex, which is the minimum or maximum point of the parabola.
- ๐ช Symmetry: The parabola is symmetric about a vertical line passing through the vertex (axis of symmetry).
๐งช What is an Exponential Function?
An exponential function is a function where the independent variable appears in the exponent. The general form of an exponential function is:
$f(x) = a^x$,
where $a$ is a constant called the base, and $a > 0$ and $a \neq 1$.
- ๐ฑ Growth/Decay: It represents exponential growth if $a > 1$ and exponential decay if $0 < a < 1$.
- asymptote Asymptote: It has a horizontal asymptote, which the graph approaches but never touches.
- ๐ Rate of Change: The rate of change increases rapidly (in growth) or decreases rapidly (in decay).
๐ Exponential vs. Quadratic Functions: Side-by-Side Comparison
| Feature |
Quadratic Function |
Exponential Function |
| General Form |
$f(x) = ax^2 + bx + c$ |
$f(x) = a^x$ |
| Graph Shape |
Parabola (U-shaped) |
Curve that increases or decreases rapidly |
| Rate of Change |
Changes at a polynomial rate |
Changes at an exponential rate (much faster for large x) |
| Asymptote |
No horizontal asymptote |
Horizontal asymptote (y = 0) |
| Turning Point |
Has a vertex (minimum or maximum) |
No turning point |
| Symmetry |
Symmetric about the vertical line through the vertex |
Not symmetric |
๐ Key Takeaways
- ๐ Identify the Form: Look at the equation. If $x$ is squared, it's likely quadratic. If $x$ is in the exponent, it's exponential.
- ๐ Observe the Growth: Exponential functions grow (or decay) *much* faster than quadratic functions for large values of $x$.
- ๐ Check for Asymptotes: Exponential functions have horizontal asymptotes, while quadratic functions do not.