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I was trying to solve this problem for the SAT math section:

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I was able to guess the answer (5) from the multiple choice options, but I'm not sure what the solution process is. The discussion on the practice site I'm using says you should calculate the two vertices (3,8 and 7,8), then figure out that t is at the midpoint of their x-values because parabolas are symmetrical, giving 5. But how would you solve this algebraically? What is the formula for the second, shifted function to calculate the intersection point(s)?

TKR
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1 Answers1

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The easy way to do it without guessing is to notice that if you draw a vertical line though $(t,v)$ (that is, the line $x = t$) then the combined figure of two parabolas is symmetric around that line. In other words, the new parabola is a mirror image of the original one reflected around the line $x = t.$

The original parabola's vertex has $x = 3,$ the new parabola's vertex is shifted $4$ units to the right, so it's at $x = 7.$ The line $x = t$ is exactly halfway between the two vertex points, so $t = 5.$

The purely algebraic way (without thinking at all about the shapes in the problem) is, as already mentioned in a comment, is to solve for $x$ in $-(x-3)^2+8=-(x-7)^2+8.$ Expand the binomials on both sides; the $x^2$ terms then cancel out and you're left with an easily-solved equation in $x.$

David K
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