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.

# Finding the vertex of a parabola in standard form

Sal rewrites the equation y=-5x^2-20x+15 in vertex form (by completing the square) in order to identify the vertex of the corresponding parabola. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• why is it that to find a vertex you must do -b/2a? is there a separate video on it?
• A parabola is defined as
𝑦 = 𝑎𝑥² + 𝑏𝑥 + 𝑐 for 𝑎 ≠ 0

By factoring out 𝑎 and completing the square, we get
𝑦 = 𝑎(𝑥² + (𝑏 ∕ 𝑎)𝑥) + 𝑐 =
= 𝑎(𝑥 + 𝑏 ∕ (2𝑎))² + 𝑐 − 𝑏² ∕ (4𝑎)

With ℎ = −𝑏 ∕ (2𝑎) and 𝑘 = 𝑐 − 𝑏² ∕ (4𝑎) we get
𝑦 = 𝑎(𝑥 − ℎ)² + 𝑘

(𝑥 − ℎ)² ≥ 0 for all 𝑥
So the parabola will have a vertex when (𝑥 − ℎ)² = 0 ⇔ 𝑥 = ℎ ⇒ 𝑦 = 𝑘

𝑎 > 0 ⇒ (ℎ, 𝑘) is the minimum point.
𝑎 < 0 ⇒ (ℎ, 𝑘) is the maximum point.
• Is there a video about vertex form?
• Not specifically, from the looks of things. When Sal gets into talking about graphing quadratic equations he talks about how to calculate the vertex. On the other hand, there are several exercises in the practice section about vertex form, so the hints there give a good sense of how to proceed.
• In which video do they teach about formula -b/2a
• Why does x+4 have to = 0?
• Because then you will have a y coordinate for a given x. When x-4 = 0 (i.e. when x =4) you are left with just y=21 in the equation: because
4-4=0
0^2=0
-3(0)=0
This leaves the equation looking like y=0+21
Then you know that when x=4 that y=21. Then you have solved for x and y.
If you want to think about it a different way you could use y=f(x). Then f(4)=21. Some people might find the f(x) way easier to understand.
• @ he mentions it's mentioned in multiple videos but I made a search and watched a few videos trying to find it in vain. Would someone kindly reply with URLs or page titles of the videos he referred to?
• I don't know there those videos are, but I think is quite easy to realize it. See: the x coordinate of the vertex is the average point between the two roots of the quadratic (the two points where the graph of the parabola intersects with the horizonal axis. So if we know that the formula for one of those roots, as per the quadratic formula, is -b+square root of -bsquared-4ac, all divided by 2a, and the formula for the other root i the same but with a minus sign on the numerator instead (this is, -b - squareroot of bsquared -4ac, all divided by 2a). Then the average between those two roots is obtained by adding those two formulas and dividing by two.

So: x coordinate of the vertex as an average point between the two roots->

((-b+sqrt(bsquared-4ac))/2a + (-b-sqrt(bsquared-4ac))/2a) / 2 =

(-b/2a + (sqrt(bsquared-4ac))/2a -b/2a -(sqrt(bsquared-4ac))/2a) / 2=

(2*-b/2a) /2 =
-b/2a

See how sqrt(bsquared-4ac)/2a and -sqrt(bsquared-4ac)/21 get cancelled out of the expression.

So the key seems to be
1) the x coordinate of the vertex is equal to the average point between the two roots of the parabola -the points where the parabola intersects the horizontal axis

2) the roots of the parabola can be found via the quadratic formula.

So applying the arithmetic average formula (a+b)/2 where a is -b+sqrt(bsquared-4ac)/2a and b is -b-sqrt(bsquared-4ac)/a gives -b/2a as solution for x coordinate of vertex. This formula also works if the parabola has only one root. In that case, the vertex would lie on the horizontal axis.-
• At how does Sal get x=4? Wouldn't the expression -3(x-4)^2 have to equal - 21 for the whole equation to equal zero?
• You want that term to be equal to zero and to do that x has to equal 4 because (4-4)^2 is equal to zero.
• Why is x vertex equal to -b/2a ?
• This video is not about the equation y=-3x^2+24x-27
It is about completing the square to solve 4x^2+40x-300=0
Can anyone help me?
The transcript is going but it is different words!
• Can someone explain why the x-vertex formula is -b/2a?
• We know we can find the x-intercepts of the parabola by using the quadratic formula.
1st x-intercept: x = [-b + sqrt(b^2-4ac)]/(2a)
2nd x-intercept: x = [-b - sqrt(b^2-4ac)]/(2a)

The x-value of the vertex is located midway between these 2 points. If you average the two x-intercepts, you get their midpoint. This means you add the 2 points then divide by 2.
ADD 1st: [-b + sqrt(b^2-4ac)]/(2a) + [-b - sqrt(b^2-4ac)]/(2a)
-- Notice, we already have a common denominator, so we add the numerators.
-- Also notice, the square roots add to 0 because one is positive and the other negative.
-- This leaves: (-b-b)/(2a) = (-2b)/(2a) = -b/a

DIVIDE BY 2: -b/a divided by 2 = -b/a * 1/2 = -b/(2a)

Hope this helps.
• How can we find the domain and range after compeleting the square form?
• The Domain of a function is the group of all the x values allowed when calculating the expression.
In this exercise all x values can be used so the domain is the group of all the Real numbers.
Examples to functions that would limit the domain would contain operations like:
Division - Because division by 0 is not allowed
Square root - Because Square root of a negative number is not a real number
As you can see there are no such operations in this exercise.
The Range of a function is the group of all the y values that result from calculating the function for all the x values allowed (the Domain).
As Sal explains in the last part of the video when you bring the parabola to its vertex form it is easier to see the Range.
The free coefficient, i.e., the C in the video, is either the minimum or the maximum point of the Range.
The sign of the leading coefficient, i.e., the A in the video, determines whether it is the minimum or the maximum.
If A>0 the parabola open upwards (we call it smiling :-) and all other values of y will be greater than C, i.e., C is minimum and the Range is y>=C
If A<0 the parabola open downwards (we call it weeping :-) and all other values of y will be smaller than C, i.e., C is maximum and the Range is y<=C
In this exercise A is (-3) and it is negative, so 21 is the maximum and the Range is y<=21
Hope it helps :-)