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# 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 does the quadratic equation have to equal 0?
and why does the root of quadratic equation need y to equal 0?
• Anything times 0 will equal 0 (1x0=0;2x0=0;3x0=0;4x0=0 etc) therefore if (x-5)(x+3) = 0, either x-5 = 0 or x+3=0, therefore either x=5 or x=-3, but if (x-5)(x+3) = 15; x can equal an infinite number of values, as long as it equals 15, therefore, one cannot definitely say what the value of x is, unless the entire equation equals 0
• 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.
• Why is x vertex equal to -b/2a ?
• 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.
• 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 :-)