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Quadratic Functions

Characteristics of Quadratic Functions

Shape of Quadratic Function Graphs

The graph of a quadratic function always forms a parabola. This parabola can open upward or downward, depending on the value of the coefficient $a$ .

Influence of the Leading Coefficient on Graph Shape

When the Leading Coefficient Is Positive

Characteristics of Quadratic FunctionsLearn the key characteristics of quadratic functions, including vertex, axis of symmetry, and intercepts, through examples.

On this page

Shape of Quadratic Function Graphs
Influence of the Leading Coefficient on Graph Shape
Important Characteristics of Quadratic Functions
Steps to Graph a Quadratic Function
Drawing Quadratic Function Graphs
- Upward-Opening Parabola Example
- Downward-Opening Parabola Example
Table of Quadratic Function Graph Shapes

Comments

If $a > 0$ , the graph of the quadratic function will open upward. This means the graph has a minimum point.

Examples of functions with $a > 0$ :

$f(x) = x^2$ (the simplest function with $a = 1$ )
$f(x) = 2x^2 + 1$ (example with $a = 2$ )

Quadratic Function Graph with

a > 0

Graph opens upward and has a minimum point.

When the Leading Coefficient Is Negative

If $a < 0$ , the graph of the quadratic function will open downward. This means the graph has a maximum point.

Examples of functions with $a < 0$ :

$f(x) = -x^2$ with $a = -1$
$f(x) = -3x^2 - 12x - 15$ with $a = -3$

Quadratic Function Graph with

a < 0

Graph opens downward and has a maximum point.

Why the Leading Coefficient Cannot Be Zero

When $a = 0$ , the function form becomes $f(x) = bx + c$ . This is no longer a quadratic function, but a linear function. A quadratic function must have $a \neq 0$ so that the highest power of the variable $x$ is $2$ .

Important Characteristics of Quadratic Functions

Vertex

The vertex is the highest point (if $a < 0$ ) or the lowest point (if $a > 0$ ) on the graph. The coordinates of the vertex are expressed as $(-\frac{b}{2a}, f(-\frac{b}{2a}))$ .

Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two symmetrical parts. The equation of the axis of symmetry is $x = -\frac{b}{2a}$ .

Vertical Intercept

The $y$ -intercept is obtained when $x = 0$ . Its value is $f(0) = c$ .

Horizontal Intercepts

The x-intercepts are obtained when $f(x) = 0$ , i.e., when $ax^2 + bx + c = 0$ . The solutions can be found using the formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Steps to Graph a Quadratic Function

Determine whether the parabola opens upward ( $a > 0$ ) or downward ( $a < 0$ ).
Calculate the coordinates of the vertex $(-\frac{b}{2a}, f(-\frac{b}{2a}))$ .
Calculate the $y$ -intercept: $(0, c)$ .
Calculate the x-intercepts (if any).
Choose several other $x$ values and calculate their corresponding $y$ values.
Plot all points in the coordinate system.
Connect the points with a parabolic curve.

Drawing Quadratic Function Graphs

Upward-Opening Parabola Example

Let's graph the function $f(x) = x^2 - 2x - 3$ :

Coefficient $a = 1 > 0$ , so the parabola opens upward.
Vertex:
$x = -\frac{b}{2a} = -\frac{-2}{2 \cdot 1} = 1$
So the vertex is at $(1, -4)$ .
$y$ -intercept:

$f(0) = 0^2 - 2 \cdot 0 - 3 = -3$
X-intercepts: $f(x) = 0$ or $x^2 - 2x - 3 = 0$

Using the quadratic formula:
Let's calculate some additional points:
$f(-2) = (-2)^2 - 2 \cdot (-2) - 3 = 4 + 4 - 3 = 5$

Graph of

f(x) = x^2 - 2x - 3

Parabola opens upward with vertex at

(1, -4)

and x-intercepts at

(-1, 0)

and

(3, 0)

.

Downward-Opening Parabola Example

Let's graph the function $f(x) = -x^2$ :

Coefficient $a = -1 < 0$ , so the parabola opens downward.
Vertex:
$x = -\frac{b}{2a} = -\frac{0}{2 \cdot (-1)} = 0$
So the vertex is at $(0, 0)$ .
$(3, 0)$ -intercept:

$f(0) = 0$

So the $y$ -intercept is at $(0, 0)$ .
X-intercepts: $f(x) = 0$ or $-x^2 = 0$
$x^2 = 0$
Let's calculate some additional points:
$f(-2) = -((-2)^2) = -4$

Graph of

f(x) = -x²

Parabola opens downward with vertex at

(0, 0)

.

Table of Quadratic Function Graph Shapes

Quadratic Function	Graph Shape
$a > 0$	Parabola opens upward, has a minimum point
$a < 0$	Parabola opens downward, has a maximum point

Types of Quadratic Equation Roots

Constructing Quadratic Functions

So the $y$ -intercept is at $(0, -3)$ .

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{2 \pm \sqrt{4 + 12}}{2} = \frac{2 \pm \sqrt{16}}{2} = \frac{2 \pm 4}{2}

x = \frac{2 + 4}{2} = 3 \text{ or} x = \frac{2 - 4}{2} = -1

So the x-intercepts are at $(-1, 0)$ and $(3, 0)$ .

f(2) = 2^2 - 2 \cdot 2 - 3 = 4 - 4 - 3 = -3

x = 0

So the $x$ -intercept is at $(0, 0)$ .

f(1) = -(1^2) = -1

f(2) = -(2^2) = -4