1. ## Help with Parabola's

Thanks in advance for the help!

So the online lesson I'm taking didn't explain how to solve parabola's in equation form very well, and didn't explain how to graph a parabola at all, so I was hoping I could get some detailed explanations on here if it's not too much trouble!

Alright, so they give me a table, and they give me the equation:

y = 3(x - 5)(x - 17)

Then they ask for five things, the Zeros, the Axis of Symmetry, the Vertex, if its a Maximum or minimum parabola, and the Maximum or minimum value.

In my mind, the zeros would be -5 and -17 in this equation, but they say that the zeros are 5 and 17. Also, in other equations, for example if that equation was "y = 3(x + 5)(x + 17), then they say the zeros would be -5 and -17. That doesn't make sense to me, so an explanation here would be really nice.

I understand how to get the axis of symmetry, just take the zeros, add them together and divide by 2.

They say the vertex is (11, -108) but I don't understand how they get -108.

I don't understand why its a minimum since the thing with the 0's is confusing me. If the zero's were negative numbers then I would understand why its a minimum, but they're saying the zeros are positives, so in my head, that would make it a maximum.

I don't really understand how I would turn this equation into a graph either, I don't know how I would solve for "x".

I don't need the answers, what I need is to understand how to get to the answers, so anyone who could explain it nicely would be very much appreciated

2. ## Re: Help with Parabola's

To find the zeroes, which are the x-intercepts, it's where y = 0. To solve nonlinear equations, a good method is to make use of the Null Factor Law, which states that if two things multiply to give 0, then at least one of those things must be 0.

So in your case, you have \displaystyle \displaystyle \begin{align*} 0 = 3(x - 5)(x - 17) \end{align*}. You have three things multiplying to give 0, namely 3, (x - 5) and (x - 17), so that means either x - 5 = 0 or x - 17 = 0 (as obviously 3 is not 0). Therefore x = 5 or x = 17.

3. ## Re: Help with Parabola's

Thanks a bunch for the help with the zeros, that really clears that part up!

If you could also explain how to get the -108 in the vertex I think I would understand enough to do all of the other questions

4. ## Re: Help with Parabola's

Your axis of symmetry gives the x value of the turning point. Substitute it into your equation to find the corresponding y value.

5. ## Re: Help with Parabola's

Ahh, so (11 - 5) x (11 - 17) then multiplied by 3 gives -108.

Well damn, it always seems so easy once you understand how it works, if only my mind thought of it that way to begin with.

Thanks again!

6. ## Re: Help with Parabola's

You could also do this (a little simpler theory, a little more calculation) by multiplying the factors out- $\displaystyle y= 3(x- 5)(x- 17)= 3(x^2- 22x+ 85)$- and "complete the square". You should know that $\displaystyle (x- a)^2= x^2- 2ax+ a^2$. Comparing that to $\displaystyle x^2- 22x+ 85$ (the "3" is not important yet) we see that we have "22x" instead of "2ax" so we can take a= 11. In that case $\displaystyle a^2= 11^2= 121$. That is, $\displaystyle (x- 11)^2= x^2- 22x+ 121$. We have $\displaystyle x^2- 22x+ 85$ but we also know that 85= 121- 36 so we can write that as $\displaystyle x^2- 22x+ 121- 36= (x- 11)^2- 36$ and so $\displaystyle y= 3((x- 11)^2- 36)$.

Now we can see that when x= 11, x- 11= 0 so [tex]y= 3(0^2- 36)= -108. And if x is NOT 22, $\displaystyle (x- 11)^2$ is positive so that y is larger than -108. The lowest point of the parabola, the vertex, is at (11, -108).

7. ## Re: Help with Parabola's

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