At what points do the graphs of [tex]y = 2x + 1[/tex] and [tex]y = -(x - 1)^2 + 3[/tex] intersect?

The question is basically asking when do the two equations equal the same thing 2x +1 = -(x-1)^2+3
move the x's all on one side, and the numbers on the other 2x + (x - 1)^2 = +2
multiply out the brackets: (x - 1)^2 (x - 1)(x -1) x^2 -2x +1
put that into the equation instead of the brackets: 2x +x^2 -2x +1 = +2
keep all the x's on one side again: 2x +x^2 -2x = 1
The 2x's cancel out, so: x^2 = +1
square root: x = √1 = 1
if x = 1 then put that into the equations to find the value of y when they cross: y=2x+1 y=2(1)+1 y= 2+1 = 3 y = 3
y= -(x-1)^2 +3 y = -((1) -1)^2 +3 = 3 y = 3
so the point where they cross is (1,3). I think. not 100% sure

Answered by jsef | 2024-06-10

The graphs of the equations intersect at the points (1, 3) and (-1, -1). To find these points, we set the equations equal to each other, solved for x, and then substituted back to find y. Therefore, the solution involves solving and substituting values to find the intersection points.
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Answered by jsef | 2024-09-05