I want to learn about quadratic functions, relations, models for up to grade 10/11

Welcome! I am excited to help you master quadratic functions, relations, and models. We will go step by step to build a solid foundation.

Before we dive into the fun stuff, I have one quick question to make sure we tailor this session perfectly to your current math level.

I have covered basic algebra recently, but quadratics are new.

It is great to have you here! Since you already have a solid handle on basic algebra, you are in a perfect position to tackle quadratics.

Let's kick things off with our roadmap for this session so you can see exactly where we are headed.

Your path

Graphing Quadratic Functions

Solving Quadratic Equations

Forms of Quadratics

Real-world Models

What Makes a Function Quadratic?

In algebra, you are probably used to working with linear equations like

y = 2x + 3

. In those equations, the independent variable

x

has an exponent of

1

(even if we do not write it out). A is different because the highest exponent (often called the degree) on any variable is exactly

2

. That single change completely transforms how the math behaves and what its graph looks like.

To spot a quadratic, look at the exponents. For example:

$y = x^2 - 4$ is quadratic because the highest power of $x$ is $2$ .
$y = 3x^2 + 2x - 5$ is also quadratic.
$y = x^3 + x^2$ is not quadratic, because the highest exponent is $3$ (making it cubic).

Continue to standard form: y = ax^2 + bx + c

You've got it! Now that you've got standard form down, we are going to explore what a quadratic function actually looks like on a graph.

The Parabola

While linear equations draw straight lines, a quadratic function always curves into a symmetric U-shape (or an upside-down U) called a . This shape shows up everywhere in the physical world, from the path of a kicked soccer ball to the curve of a suspension bridge cable.

A standard parabola centered on the y-axis, opening upwards on a coordinate plane with its vertex at the origin and its vertical axis of symmetry highlighted.

When you look at a parabola, there are two key properties to keep in mind:

The Vertex: This is the turning point of the curve. It's either the absolute lowest point (a minimum) or the absolute highest point (a maximum).
The Axis of Symmetry: This is an imaginary vertical line that cuts the parabola perfectly in half. If you folded the graph along this line, both sides of the parabola would match up perfectly.

Continue to calculating the axis of symmetry and vertex

Let's learn how to find these vital properties mathematically directly from the standard form equation.

Finding the Axis of Symmetry

When your quadratic function is written in standard form,

y = ax^2 + bx + c

, calculating the vertical is surprisingly straightforward. Since a parabola is perfectly balanced on its left and right sides, its center line always passes directly through the vertex. We find the exact equation of this vertical line using a simple formula built from the coefficients

a

and

b

x = -\frac{b}{2a}

Calculating the Vertex

Now, what about the vertex? Because the vertex sits directly on the axis of symmetry, its x-coordinate is exactly the value we just calculated:

x = -\frac{b}{2a}

. To find the corresponding y-coordinate, you simply plug that x-value back into your original quadratic equation and solve. This gives you the coordinates of the of your graph.

Let's walk through a concrete example. Imagine we have the quadratic function:

y = 2x^2 - 8x + 3

First, identify your coefficients:

$a = 2$
$b = -8$
$c = 3$

Step 1: Find the axis of symmetry using our formula:

x = -\frac{-8}{2(2)}

x = -\frac{-8}{4}

x = 2

So, our axis of symmetry is the vertical line

x = 2

. Step 2: Find the vertex coordinate by plugging

x = 2

back into our function:

y = 2(2)^2 - 8(2) + 3

y = 2(4) - 16 + 3

y = 8 - 16 + 3

y = -5

Combining these steps, the vertex of this parabola is located at the coordinates

(2, -5)