Sketch the curve y = x^2 - 6x + 5, identifying roots and minima/maxima.
Remeber the formula: (a - b)2 = a2 - 2ab + b2. Notice that y = x2 - 23x + 5, so we want to write this using (x - 3)2 = x2 - 23x* + 9. Taking 4 from both sides gives: (x - 3)2 - 4 = x2 - 6x + 5 = y.
We need some simple facts about graphs: (1) y = x2 is a parabola (U shaped); (2) if we replace x wih x - 3 we move the graph to the right by 3; (3) if we add -4 to y, the graph moves down by 4.
To find minima: notice that (x - 3)2 is always positive or 0, so (x - 3)2 + -4 >= -4. If x is not 3, then (x - 3)2 > 0, so y > -4; but if x = 3, we have y = -4, so -4 is the smallest value of y (i.e. a minimum) at (3, -4).
To find roots, we can solve the quadratic y = 0:
(x - 3)2 - 4 = 0 <=> (x - 3)2 = 4 <=> x - 3 = 2 or x - 3 = -2 <=> x = 5 or x = 1.
(Rememer that x2 = a2 has two solution: x = a and x = -a.) With this it should be easy to sketch the curve!
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