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Let f(x)=3x^2 - 4x - x^2 + 7x + 8. Find the constant k such that f(x)=f(k-x) for all real numbers x.

 Apr 30, 2022
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First, simplify f(x) to 2x²+3x+8.

f(k-x) just reflects f(x) over the y-axis and then translates it to the left by k. The y-intercept of the parent funciton and of f(k-x) would remain the same: 8. The y-intercept would be on the right side of the vertex in f(x) and on the left side in f(-x), so f(-x) has to be shifted left until thegraphs are the same.

2x²+3x+8=8

2x²+3x=0

x(2x+3)=0

x=0 or x=-1.5

Because x=0 would mean the y-inercept, which we already know, the other place the parent y=8 would be at (-1.5, 8), which means k would have to be -1.5 to shift f(k-x) to f(x).

 

I know this isn't a well-worded at all, but hopefully it helps. I suggest you spend some time on a graphing application looking at how what changes to a function causes what transformations.

 Apr 30, 2022

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