Lesson 1: Key Features of Graphs of a Quadratic Function

Property

A quadratic function in the form $f(x) = ax^2$, where $a$ is a real number and $a \neq 0$, is the simplest type of quadratic function.

We call the graph of a quadratic function a parabola.

Property

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

• The graph has either a highest point or a lowest point, called the vertex.
• The parabola is symmetric about a vertical line, called the axis of symmetry, that runs through the vertex.
• The parabola opens either upward or downward, depending on the sign of the coefficient $a$.
• The vertex represents either the minimum value (when opening upward) or maximum value (when opening downward) of the function.

Property

The quadratic parent function $f(x) = x^2$ has its axis of symmetry at $x = 0$, which is the vertical line that divides the parabola into two mirror-image halves.

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.