Answer:
![\displaystyle \large{y=x^2-2x-3}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clarge%7By%3Dx%5E2-2x-3%7D)
Step-by-step explanation:
The given points are (-1,0) and (3,0) to find an equation of parabola. Notice that they say or the given points are x-intercepts, which means they make it easier for us to find an equation of parabola using these x-intercepts.
Quick Note:
- x-intercepts are simply the roots or solutions of quadratic equation, so if our root is x = -2 then we can write as (-2,0) and write back to x+2=0.
So what we are going to do is to write the roots’ equation back as ![\displaystyle \large{x+\alpha =0}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clarge%7Bx%2B%5Calpha%20%3D0%7D)
Our points or roots are given, therefore:
If x = -1 then x+1=0
If x = 3 then x-3=0
Then multiply x+1 with x-3 because if (x-1)(x+3)=0 then x-1=0 or x+3=0:
![\displaystyle \large{(x+1)(x-3)=0}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clarge%7B%28x%2B1%29%28x-3%29%3D0%7D)
Convert 0 to y since we want to find a function:
![\displaystyle \large{(x+1)(x-3)=y}\\\displaystyle \large{y=(x+1)(x-3)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clarge%7B%28x%2B1%29%28x-3%29%3Dy%7D%5C%5C%5Cdisplaystyle%20%5Clarge%7By%3D%28x%2B1%29%28x-3%29%7D)
Expand the expressions in and simplify to standard form:
![\displaystyle \large{y=x^2-3x+x-3}\\\displaystyle \large{y=x^2-2x-3}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clarge%7By%3Dx%5E2-3x%2Bx-3%7D%5C%5C%5Cdisplaystyle%20%5Clarge%7By%3Dx%5E2-2x-3%7D)
Therefore, the equation is ![\displaystyle \large{y=x^2-2x-3}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clarge%7By%3Dx%5E2-2x-3%7D)
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Summary
If we are given the points
and
with wideness of the graph or a-value = 1 then the parabola equation is:
![\displaystyle \large{y=(x-\alpha )(x-\beta )}\\\displaystyle \large{y=x^2-\beta x-\alpha x+\alpha \beta }\\\displaystyle \large{y=x^2-(\beta +\alpha )x+\alpha \beta }](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clarge%7By%3D%28x-%5Calpha%20%29%28x-%5Cbeta%20%29%7D%5C%5C%5Cdisplaystyle%20%5Clarge%7By%3Dx%5E2-%5Cbeta%20x-%5Calpha%20x%2B%5Calpha%20%5Cbeta%20%7D%5C%5C%5Cdisplaystyle%20%5Clarge%7By%3Dx%5E2-%28%5Cbeta%20%2B%5Calpha%20%29x%2B%5Calpha%20%5Cbeta%20%7D)
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Others
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