If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Forms & features of quadratic functions

Different forms of quadratic functions reveal different features of those functions. Here, Sal rewrites f(x)=x²-5x+6 in factored form to reveal its zeros and in vertex form to reveal its vertex. Created by Sal Khan.

Want to join the conversation?

• The vertex from what I have understood from the video is the lowest point in a parabola. But what does the roots imply?
• The roots are the x-intercepts, where the parabola crosses the x-axis. If the parabola opens up and it's vertex is below the x-axis then it crosses the x-axis in two places and has two (real) roots. If the vertex is on the x-axis then the parabola has one root. If the vertex is above the x-axis (and the parabola opens up) then that parabola has no real roots... it still has roots, but they are complex numbers.
• Good question!
The word `quadratic` refers to the degree of a polynomial such as x² - 4x + 3
To be quadratic, the highest power of any term must be 2 (the x is squared). If there is `no equals sign`, but it has a quadratic term, then it is a `quadratic expression`.
x² - x - 5 is a `quadratic expression.`
So are the following:
a² + 8a - 6
g² - 16
If there is an equals sign, we call it a `quadratic equation`.
An example is
x² - 4x + 3 = 0
Another example of `quadratic equation` is
16t² - 8t = 3
and,
x = y²
and
1 + 4/5x - x² = 0
If we are defining how two variables are related, we can be talking about a `function`. There are limits on what we can call a function, though, because there has to be one unique value for the dependent variable for any value of the independent variable in order to be a function.
y = x² - 8x + 15 is a `quadratic function` that describes every point on a parabola
f(x) = x² - 4x + 3 is also a `quadratic function` that describes every point on another parabola
The next `quadratic equation` looks similar, but is not a quadratic function: it is not a function at all, in fact.
x = y² - 8y - 20
It describes a parabola that is lying on its side on a coordinate grid, but most values of x for this parabola that have any value at all will have two different values of y, so it cannot be a function. It would still be classified as a `quadratic equation`
• Is there a easy trick for knowing if whether a parabola is a mininum or a maxinum?
• If you look at the first term before the parenthesis, there would be a number. If its negative, it has a maximum. If its positive, then it has a minimum.
• I obtained the vertex by solving the original equation over the average of the roots of x and got the same value - this is to say that I added the roots and divided by 2 then used the result as x. Is this a legitimate way to find the vertex for all cases? Does anyone know why this works?
• Yes, since a parabola is symmetric about the line of symmetry, you can find it by finding the mid point of the two zeros; however, unless you also need to find the roots for some other reason, it is certainly not the most efficient way to find the vertex.
• I have a quadratic equation that I cannot figure out.
y=-x^2-4x-3
I know the parabola points downward because a is a negative, but I cannot seem to find the axis of symmetry and the vertex. I have the y-intercept, -3.
I keep getting 2 for the axis and -7 for the vertex. None of these answers fit on the graph. Could someone explain this to me, please? Thank you.
• Hi apandafamily. You are correct that the y-intercept is -3. However, the rest of your solution is incorrect so let's take a look at how to solve this. First, I like to factor out the negative sign so I can more easily find the roots and minimize errors in my calculation. Let's rewrite our equation.
y=-x²-4x-3 then becomes y=-(x²+4x+3). Now when we factor, this becomes
y=-(x+3)(x+1). Therefore, the roots are 0=-(x+1) so x=-1 and 0=-(x+3) so x=-3. Now take a look at the roots on a number line. The axis of symmetry will have to lie between -3 and -1 so it will be -2. If you substitute -2 into either the original equation or even our modified equation, y=-1*(x²+4x+3), you'll get y=-1*((-2)²+4*(-2)+3)=-1*(4-8+3)=-1*-1=1.
So in summary, the roots of the polynomial are x=-1 and x=-3 with the vertex at (-2,1).
• Is there a formula for finding the vertex of a parabola? Is there a video on that topic?
• Cannot we get the vertex by taking the average of x intercepts? *like 3+2/2=5/2(same answer shown in the video 5/2)* and the minimum value by plugging in 5/2 to the equation ?which is also gonna give -1/4
i think we can 🤔
• Yes, you're absolutely right that you can. Sometimes though, using complete the square would be faster. And it's good to know why we have vertex form(to know the vertex!)
Factored form: Best for x-intercepts
Vertex form: Best for vertex
Standard form & quadratic formula: Can work out y-intercept
Though you can absolutely work out everything just using one form, it's good to be versatile and choose the fastest or easiest way :)
• Why is y = f(x) and not x = f(x)?
• f(x) is read as f of x, and it means what is the value of the function at x, this is the y value, so y=f(x).
• Can someone explain to me what Sal just did? I'm so confused.
• Basically a quadratic function can be rewritten in various forms.

1. f(x) = ax² + bx + c

We can determine the sum and product of the roots of x easily.
Sum = -b / a
Product = c / a

2. f(x) = a(x - b)(x - c)

We can determine the roots of x easily.
x = b or x = c

3. f(x) = a(x - b)² + c

We can determine the vertex easily.
Vertex: (b, c)

TIP: You can determine the concavity by looking at the sign of a.