Lesson 1: Graphing f (x) = ax²

Property

A basic quadratic function has the form

where $a$ is a real number and $a \neq 0$. The graph of this function is a parabola with vertex at the origin $(0, 0)$.

Examples

Property

The graph of the quadratic function $f(x) = ax^2$ is called a parabola. All parabolas of this form share certain key features:

• The graph has either a highest point or a lowest point, called the vertex, which is always at the origin $(0, 0)$.
• The parabola is symmetric about the $y$-axis (the axis of symmetry).
• The parabola has a $y$-intercept at $(0, 0)$.
• If $a > 0$, the parabola opens upward; if $a < 0$, the parabola opens downward.

Examples

Property

• The parabola opens upward if $a > 0$.
• The parabola opens downward if $a < 0$.
• The magnitude of $a$ determines how wide or narrow the parabola is.
• The vertex, the $x$-intercepts, and the $y$-intercept all coincide at the origin.

Examples

• The graph of $y = 4x^2$ opens upward and is narrower than the basic parabola $y=x^2$. It passes through the points $(-1, 4)$ and $(1, 4)$.
• The graph of $y = -\frac{1}{3}x^2$ opens downward and is wider than the basic parabola. It passes through the points $(-3, -3)$ and $(3, -3)$.
• The graph of $y = 0.25x^2$ opens upward and is wider than the basic parabola. It passes through the points $(-2, 1)$ and $(2, 1).

Explanation

The coefficient 'a' acts like a stretch factor that controls the parabola's direction and width. A positive 'a' makes it open up, while a negative 'a' flips it upside down. A larger absolute value of 'a' creates a narrower parabola.