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.
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- why is it that to find a vertex you must do -b/2a? is there a separate video on it?(38 votes)
- 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.(35 votes)
- Is there a video about vertex form?(9 votes)
- 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.(12 votes)
- In which video do they teach about formula -b/2a(13 votes)
- This is not a derivation or proof of -b/2a, but he shows another way to get the vertex: https://www.khanacademy.org/math/algebra/quadratics/features-of-quadratic-functions/v/quadratic-functions-2
There are some answers to the derivation of -b/2a here:
- Why does x+4 have to = 0?(9 votes)
- 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
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.(14 votes)
- Why is x vertex equal to -b/2a ?(4 votes)
- This is not a derivation or proof of " -b/2a", but he shows another way to get the vertex: https://www.khanacademy.org/math/algebra/quadratics/features-of-quadratic-functions/v/quadratic-functions-2
There are some answers to the derivation of -b/2a here:
- At3:38how does Sal get x=4? Wouldn't the expression -3(x-4)^2 have to equal - 21 for the whole equation to equal zero?(5 votes)
- 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.(3 votes)
- 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!(3 votes)
- How can we find the domain and range after compeleting the square form?(0 votes)
- 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 :-)(14 votes)
- How are you today?(2 votes)
Can someone please explain what Sal meant when he said the graph would be at minimum 3? Thank you!(2 votes)
I have an equation right here. It's a second degree equation. It's a quadratic. And I know its graph is going to be a parabola. Just as a review, that means it looks something like this or it looks something like that. Because the coefficient on the x squared term here is positive, I know it's going to be an upward opening parabola. And I am curious about the vertex of this parabola. And if I have an upward opening parabola, the vertex is going to be the minimum point. If I had a downward opening parabola, then the vertex would be the maximum point. So I'm really trying to find the x value. I don't know actually where this does intersect the x-axis or if it does it all. But I want to find the x value where this function takes on a minimum value. Now, there's many ways to find a vertex. Probably the easiest, there's a formula for it. And we talk about where that comes from in multiple videos, where the vertex of a parabola or the x-coordinate of the vertex of the parabola. So the x-coordinate of the vertex is just equal to negative b over 2a. And the negative b, you're just talking about the coefficient, or b is the coefficient on the first degree term, is on the coefficient on the x term. And a is the coefficient on the x squared term. So this is going to be equal to b is negative 20. So it's negative 20 over 2 times 5. Well, this is going to be equal to positive 20 over 10, which is equal to 2. And so to find the y value of the vertex, we just substitute back into the equation. The y value is going to be 5 times 2 squared minus 20 times 2 plus 15, which is equal to let's see. This is 5 times 4, which is 20, minus 40, which is negative 20, plus 15 is negative 5. So just like that, we're able to figure out the coordinate. This coordinate right over here is the point 2, negative 5. Now it's not so satisfying just to plug and chug a formula like this. And we'll see where this comes from when you look at the quadratic formula. This is the first term. It's the x value that's halfway in between the roots. So that's one way to think about it. But another way to do it, and this probably will be of more lasting help for you in your life, because you might forget this formula. It's really just try to re-manipulate this equation so you can spot its minimum point. And we're going to do that by completing the square. So let me rewrite that. And what I'll do is out of these first two terms, I'll factor out a 5, because I want to complete a square here and I'm going to leave this 15 out to the right, because I'm going to have to manipulate that as well. So it is 5 times x squared minus 4x. And then I have this 15 out here. And I want to write this as a perfect square. And we just have to remind ourselves that if I have x plus a squared, that's going to be x squared plus 2ax plus a squared. So if I want to turn something that looks like this, 2ax, into a perfect square, I just have to take half of this coefficient and square it and add it right over here in order to make it look like that. So I'm going to do that right over here. So if I take half of negative 4, that's negative 2. If I square it, that is going to be positive 4. I have to be very careful here. I can't just willy nilly add a positive 4 here. I have equality here. If they were equal before adding the 4, then they're not going to be equal after adding the 4. So I have to do proper accounting here. I either have to add 4 to both sides or I should be careful. I have to add the same amount to both sides or subtract the same amount again. Now, the reason why I was careful there is I didn't just add 4 to the right hand side of the equation. Remember, the 4 is getting multiplied by 5. I have added 20 to the right hand side of the equation. So if I want to make this balance out, if I want the equality to still be true, I either have to now add 20 to y or I have to subtract 20 from the right hand side. So I'll do that. I'll subtract 20 from the right hand side. So I added 5 times 4. If you were to distribute this, you'll see that. I could have literally, up here, said hey, I'm adding 20 and I'm subtracting 20. This is the exact same thing that I did over here. If you distribute the 5, it becomes 5x squared minus 20x plus 20 plus 15 minus 20. Exactly what's up here. The whole point of this is that now I can write this in an interesting way. I could write this as y is equal to 5 times x minus 2 squared, and then 15 minus 20 is minus 5. So the whole point of this is now to be able to inspect this. When does this equation hit a minimum value? Well, we know that this term right over here is always going to be non-negative. Or we could say it's always going to be greater than or equal to 0. This whole thing is going to hit a minimum value when this term is equal to 0 or when x equals 2. When x equals 2, we're going to hit a minimum value. And when x equals 2, what happens? Well, this whole term is 0 and y is equal to negative 5. The vertex is 2, negative 5.