Lesson 84: Identifying Quadratic Functions

New Concept

The standard form of a quadratic function is $f(x) = ax^2 + bx + c$, where $a, b,$ and $c$ are real numbers and $a \neq 0$.

What’s next

Next, you’ll learn to identify these functions, graph their characteristic 'parabola' shape, and predict their orientation from the equation alone.

Property

A quadratic function is a function that can be written in the form $f(x) = ax^2 + bx + c$, where $a$ is not equal to 0. It is a polynomial with a degree of exactly 2.

Explanation

Think of the $x^2$ term as the superstar of the function! The word "quadratic" comes from "quadratus," meaning square. If a function doesn't have an $x^2$ term as its highest power, it simply can't join the quadratic club. It’s the key ingredient for that U-shaped graph.

Examples

$y + 5x = 2x^2 - 3$ simplifies to $y = 2x^2 - 5x - 3$, which is a quadratic function.
$y = 10 + 3x$ is not a quadratic function because it lacks an $x^2$ term.
$y = -5x^3 + x^2$ is not a quadratic function because its highest power is 3, making it cubic.

Let's see if we can unmask this equation to find the quadratic function in disguise. This relates to our first key idea: identifying if a function is quadratic.

Example Problem

Determine whether the function $y + 5x = 2x^2 - 8$ is a quadratic function.

Property

The standard form of a quadratic function is $f(x) = ax^2 + bx + c$, where $a, b,$ and $c$ are real numbers and $a \neq 0$.

Explanation

Writing functions in standard form is like organizing your closet. You put the most powerful item first (the $ax^2$ term), followed by the next ($bx$), and finally the constant ($c$). This neat arrangement makes it super easy to identify the key values $a, b,$ and $c$ for solving problems.

Examples

The function $f(x) = 5x - 2x^2 + 1$ in standard form is $f(x) = -2x^2 + 5x + 1$.
The equation $y + 7x = 4x^2 - 6$ in standard form is $y = 4x^2 - 7x - 6$.
The function $f(x) = 8x^2 - 4$ is in standard form where $a=8$, $b=0$, and $c=-4$.

Property

For a quadratic function in standard form, $y = ax^2 + bx + c$: If $a < 0$, the parabola opens downward. If $a > 0$, the parabola opens upward.

Explanation

The value of 'a' is the boss of the parabola's direction! If 'a' is positive, think of a happy smile—the parabola opens up. If 'a' is negative, imagine a sad frown—the parabola opens down. Just look at the sign of 'a' to know which way it goes!

Examples

In $f(x) = 5x^2 + 2$, we see $a=5$. Since $a > 0$, the parabola opens upward.
In $f(x) = 4x - 2x^2 + 1$, first write it as $f(x) = -2x^2 + 4x + 1$. Since $a=-2$ ($a < 0$), it opens downward.
In $f(x) = -x^2 - 9$, we see $a=-1$. Since $a < 0$, the parabola opens downward.