What is the significance of the equation y=2(x-4)^2-3?

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SUMMARY

The equation y=2(x-4)^2-3 represents a parabola in vertex form, where the vertex is located at the point (4, -3). The coefficient 'a' is 2, indicating that the parabola opens upwards and is thinner than the standard parabola y=x^2. The discussion highlights the importance of recognizing the vertex form for quickly determining key characteristics of the parabola without extensive calculations. The user initially attempted to convert the equation to standard form but was confused about the process and the results.

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  • Understanding of quadratic equations and their forms, specifically vertex form and standard form.
  • Familiarity with the properties of parabolas, including vertex, direction of opening, and width.
  • Knowledge of the quadratic formula and its application in solving quadratic equations.
  • Basic algebraic manipulation skills for expanding and rearranging equations.
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  • Study the characteristics of parabolas in vertex form and how to identify their features.
  • Learn how to convert between vertex form and standard form of quadratic equations.
  • Explore the implications of the coefficient 'a' on the shape and orientation of parabolas.
  • Practice solving quadratic equations using the quadratic formula and analyzing their graphs.
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Students in introductory mathematics courses, particularly those studying algebra and quadratic functions, as well as educators looking for clear explanations of parabola properties.

DethRose
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Im In a 1st Year University math class and its been a while since i did this stuff and can't remember if I am doing this question right so any help would be much appreciated.

The question is to find all the info from the equation y=2(x-4)^2-3

I used the b+/- b^2 -4ac... equation and ended up getting 16+/- square root of 24/4.

This gives an odd number (5.22474, 2.77525) so i don't think it is right. When i apart the original equation to make into ax^2+bx+c=0 form i did 2(x^2-8x+16)-3=0 and i am not sure if that is correct either or if it should be multiplies without brackets.

Thanks for any help
 
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I haven't checked your working out, but I don't see what's wrong with the numbers it gives? You were never given an indication the numbers would be pretty.
 
The parabola you have is written in a very nice form...

y = a(x - h)^2 + k

"The question is to find all the info from the equation "... :-/

All the info? Be more specific. From this form it is easy to find the Vertex (h,k). You can also tell whether or not it will be opening up, or down. If a >0 then the parabola will open up. If a<0 then the parabola will be opening down.

Furthermore, a will tell you how thin or fat your parabola will be. If 0<|a|<1 your parabola will be fatter than the parabola y=x^2. If |a|>1 your parabola will be thinner than y = x^2.

As you can see one can tell a lot by the equation given without having to do much work. Is there further information that you need from the equation given? If so, what?
 

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