# Properties of Quadratic Functions

In this video, we are going to look at properties of quadratic functions. Quadratic functions are written in the form $y=ax^2+bx+c$. The most basic one is $y=x^2$. These functions are parabolas and are “U”-shaped. The point where it stops and changes to the other direction is known as the vertex. The axis of symmetry of the parabola passes through the vertex.

A graphing calculator is a helpful tool when it comes to dealing with quadratic functions. After graphing the function, the vertex can sometimes be seen as a point on the table. The axis of symmetry has the equation of the x-coordinate of the vertex. In the function $y=x^2+4x-2$ the table helped us determine that the vertex lies at (-2,-6).
Without a graphing calculator, we can still find this information.
The formula for the axis of symmetry is written as $x=\frac{-b}{2a}$
In the function $y=x^2+4x-2$ $a=1, b=4, c=-2$. When these values are substituted into the formula, $x=\frac{-(4)}{2(1)}$ we find that the axis of symmetry is at x=-2.
Since the vertex is a point on the graph, we can use the original function and the x-value of the axis of symmetry to solve for the point where the vertex lies.
If we substitute -2 for x in this function, we have
$y=(-2)^2+4(-2)-2$
which is simplified to
$y=4-8-2$
and finally
$y=-6$
This means that the vertex lies at the point (-2,-6), which agrees with the same point that we determined from the graphing calculator.

## Video-Lesson Transcript

Let’s go over properties of quadratic functions.

Quadratic functions have this form:

$y = ax^2 + bx + c$

and the more basic for is:

$y = x^2$

The graph looks like a letter “U” and is symmetrical on the $y$-axis. It can be anywhere in the graph or upside down.

If you look at the line, it starts from the top then goes down and stops at a point then back up again.

The point where it stops and turns the other direction is called a vertex.

Also, the axis of symmetry is through the vertex.

What about when the vertex is not on a particular tick mark?

We might not know the exact numbers.

But of course, there’s a way to solve it algebraically.

Before we get to solving it algebraically, let’s do it first on a calculator.

We have $y = x^2 + 4x - 2$

Let’s look at the graphing calculator.

It goes down then stops and goes back up again.

There’s the vertex – where it goes up and down on the axis of symmetry where it’s a reflector of the other side.

The calculator already identified the values for us.

According to the calculator, the vertex is $(-2, -6)$

Then let’s graph.

The axis of symmetry is $x = \dfrac{-b}{2a}$

Remember that the basic format for quadratic functions is $y = ax^2 + bx + c$.

So given that our line is

$y = x^2 + 4x - 2$

our values are as follows:

$a = 1$
$b = 4$
$c = -2$

Now, let’s solve for the axis of symmetry

$x = \dfrac{-b}{2a}$
$x = \dfrac{-(4)}{2(1)}$
$x = \dfrac{-4}{2}$

$x = -2$

Now, how do we find the vertex?

The vertex is a point on the graph.

So we could use the function to find out this point.

At this point, we know what is the value of $x$. So how do we find out the value of $y$?

Let’s use the function and substitute the value of $x = -2$.

$y = x^2 + 4x - 2$
$y = -2^2 + 4 (-2) - 2$
$y = 4 - 8 - 2$
$y = -6$

So our vertex coordinates are

$(-2, -6)$

Just like the answer on the calculator earlier.