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# Interpret quadratic models: Vertex form

Given a quadratic function that models a relationship, we can rewrite the function to reveal certain properties of the relationship. Created by Sal Khan.

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• Can someone quickly run me through all the tips in all the forms. Your answer would be so much help to others and me! You might get more than just 10 votes.

Vertex form
How to find vertex
Example of that equation
Etc.

Standard
How to find vertex
Example of that equation
Etc.

Factored
How to find vertex
Example of that equation
Etc. • Vertex form is a form of a quadratic equation that displays the x and y values of the vertex.
f(x)= a(x-h)^2+k.
You only need to look at the equation in order to find the vertex.
f(x)= 2(n-2)^2-10
In this case, the vertex is located at (2,-10).
Explanation: since -2 is in the parenthesis, the quadratic equation shifts 2 units to the right. Since the -10 is the constant, the equation shifts 10 units down.
Standard form is another form of a quadratic equation.
f(x) = ax^2+bx+c
To find the vertex in this form, you need to take negative b and divide it by 2a.
Example: x^2+4x+4
Since b =4 and a=1, -(4)/2(1)= -4/2 = -2.
Now that the x-coordinate of the vertex is known, you can substitute the x value in the equation.
f(x) = (-2)^2+4(-2)+4
f(x) = 4-8+4
f(x) = 0
The vertex is located at (-2,0).
The factored form of quadratic equations is basically the product of the two binomials that led to the quadratic equation. This allows you to see the x-intercepts of the quadratic.
(x+a)(x+b)
To find the vertex in this form, you must take the average of the zeroes of the equation. In order to find the zeroes, you must put the value of f(x) to zero and solve for both values of x.
(x+2)(x-3) = 0
x+2 = 0 and x-3 = 0
x=-2 and x=3
Now, take the average of the zeroes.
-2+3/2= 1/2
This means that the x value of the vertex is equal to 1/2.
Substitute the value of x into the equation.
(1/2+4/2)(1/2-6/2)
(5/2)(-5/2)
-25/4
So, the vertex is located at (1/2,-25/4)
Hope this helps, and sorry it was so long. I really needed to explain everything to avoid confusion.
• How is it that sometimes, I can just simplify the coefficients and sometimes I can't. Like, in other problems, I could just simplify the 2 out so that it could be t^2-10t but for this one I need to take out the 2 like 2(t^2-10t.. etc
Can I only simplify it down when it's in like standard form or something? • First, I factored v(t)=2t^2-20t to be 0=2t(t-10). This gave me the correct zeros (0,0) and (10,0) which I used to get the axis of symmetry (t=5) which got me the vertex, (5,-50).

I double checked using the Completing the Square method. This is where I'm a little confused and making some assumptions. I set the equation to 0. I made it 0=2t^2-20t. I know that you can only complete the square if the first term is equal to 1. I divided everything by 2 and eventually got 0=(t-5)^2-25. This also got me the correct zeros, however, I was under the impression this step in the process was the equivalent of converting the function to vertex form. It seems this is not the case. This is the assumption I'm making and I'm wondering if it's true. Completing the square and converting to vertex form are not the same process. The main distinction is that when you have the equation set to 0, you can divide everything by the leading coefficient, including zero, but when it is set to f(x) for example, you must factor out the leading coefficient(?), which, in this case, is NOT the GCF. That would be 2t, not just 2. You must factor out the leading coefficient and THEN complete the square, which leads to different numbers than if you set everything to 0 and just divided by 2. In this case it leads to v(t)=2(t-5)^2-50, not (t-5)^2-25, which was a step in my process of solving for the zeros using the Completing The Square method. Please correct anything wrong in my understanding of this. • The zeros of 𝑣(𝑡) = 2𝑡² − 20𝑡 are all the values of 𝑡 for which 𝑣(𝑡) = 0,
so to find the zeros we solve the equation 2𝑡² − 20𝑡 = 0.

At its vertex, however, we don't know what 𝑣(𝑡) is, so we can't set it equal to zero.
This means that when dividing by 2, we actually get
𝑣(𝑡)∕2 = 𝑡² − 10𝑡

After completing the square, we then have
𝑣(𝑡)∕2 = (𝑡 − 5)² − 25

Now we can multiply the 2 back to the right-hand side, which gives us
𝑣(𝑡) = 2(𝑡 − 5)² − 50 ⇒ 𝑣(5) = −50
• So - How do we find the zeros of 2t^2 - 20t?

I am completely lost now. I thought I understood how to do this but the 2 at the beginning is messing everything up for me. • At about , I understand how he zeroes out the -5, but wouldn't the -50 become -100 (2x-50)?
(1 vote) • how do you know when to use factored or vertex form when solving a equation?
(1 vote) • Factored form is best to find roots assuming that there are roots. You could then easily find axis of symmetry and by substitution find the vertex. Vertex form allows you to solve by taking the square root. If you want to find the solutions (x intercepts), factored form is easiest since it will generally give whole numbers. However, solutions are not always whole numbers, so not easily factorable.
(1 vote)
• Im extremely confused when it comes to completing the square, how are you getting these numbers?
(1 vote) • Why does it have to be a subtraction sign if the answer is positive? Like why can't you put (t+3)^2 instead of (t-3)^2 