Graphing Question: x<1, 1≤x≤2, x>2

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The function f(x) is defined piecewise with three segments: a linear function for x<1, a quadratic function for 1≤x≤2, and a linear function for x>2. The confusion arises from the nature of the quadratic segment, which produces a parabolic curve, while the other segments are linear. The graph of f(x) = x^2 is indeed a parabola, meaning the points in the interval 1≤x≤2 will be connected by a curve rather than a straight line. Therefore, the segments of the function are not all connected by straight lines; only the linear segments are straight, while the quadratic segment is curved. Understanding these distinctions is crucial for accurately graphing the piecewise function.
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The function is..

f(x) = 2x-2 if x<1
x^2 if 1 <= x <= 2
7-x if x>2

I made a table of values and graphed the points. The problem is my prof said something about not all of the points being connected by straight lines, some being connected by curves. I was wondering why this is, and for which points?
 
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What does the graph of f(x) = x2 look like?
 
It's a parabola.

..so would the line between the two points of the function f(x) = x^2 be curved and all others straight?
 
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