Lesson 2: Characteristics of Quadratic Functions

Property

For the graph of the quadratic function $f(x) = ax^2 + bx + c$, if

• $a > 0$, the parabola opens upward $\uparrow$
• $a < 0$, the parabola opens downward $\downarrow$

Examples

• For $f(x) = 5x^2 - 2x + 10$, the parabola opens upward because $a=5$, which is positive.
• For $f(x) = -2x^2 + 8x$, the parabola opens downward because $a=-2$, which is negative.
• For $f(x) = \frac{1}{3}x^2 - 1$, the parabola opens upward because $a=\frac{1}{3}$, which is positive.

Explanation

The sign of the leading coefficient, $a$, controls which way the parabola opens. Think of it this way: a positive 'a' is like a smile (opens up), while a negative 'a' is like a frown (opens down).

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.