Lesson 3: Quadratic Functions in Standard Form

Property

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

Examples

• The function $f(x) = 3x^2 + 6x - 2$ is a quadratic function where $a=3$, $b=6$, and $c=-2$.
• The function $g(x) = -x^2 + 5$ is a quadratic function where $a=-1$, $b=0$, and $c=5$.
• The function $h(x) = x^2 + 4x$ is a quadratic function where $a=1$, $b=4$, and $c=0$.

Explanation

A quadratic function creates a U-shaped graph called a parabola. Unlike linear functions that form straight lines, these functions include a squared term ($x^2$), which creates the distinctive curve. The values of $a$, $b$, and $c$ determine the parabola's shape and position.

Property

In the standard form of a quadratic function $f(x) = ax^2 + bx + c$, the constant term $c$ represents the y-intercept of the parabola. The y-intercept occurs at the point $(0, c)$.

Examples

Property

The graph of the function $f(x) = ax^2 + bx + c$ is a parabola where:

• the axis of symmetry is the vertical line $x = -\frac{b}{2a}$.
• the vertex is a point on the axis of symmetry, so its $x$-coordinate is $-\frac{b}{2a}$.
• the $y$-coordinate of the vertex is found by substituting $x = -\frac{b}{2a}$ into the quadratic equation.

Examples

• For $f(x) = x^2 - 6x + 11$, the axis of symmetry is $x = -\frac{-6}{2(1)} = 3$. The vertex is $(3, f(3))$, which is $(3, 2)$.
• For $f(x) = -2x^2 - 8x - 5$, the axis of symmetry is $x = -\frac{-8}{2(-2)} = -2$. The vertex is $(-2, f(-2))$, which is $(-2, 3)$.
• For $f(x) = 4x^2 - 8$, the axis of symmetry is $x = -\frac{0}{2(4)} = 0$. The vertex is $(0, f(0))$, which is $(0, -8)$.

Explanation

The axis of symmetry is an invisible vertical line that splits the parabola into two perfect mirror images. The vertex is the parabola's turning point (either the very bottom or very top), and it always sits right on this line.