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YOU are better than YOU think. Show yourself how:
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-/[]\- Logic chapters 1 to 5 re- appear not in sequence, as is or longer, in Volume 1A, Pattern Based Reason, Bon Appetite. Logic
Mastery Logic mastery makes the hard, easier. Logic mastery leads to better, stronger and richer comprehension. Logic mastery improves reading and writing. Logic mastery ease learning difficulties. Logic mastery gives a headstart. In sum, logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck. After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6;
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-/[]\- What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts. Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice. |
Summary and ExtensionA. Graphing the Standard Form. The quadratic y = a[(x-h)2 + k ] has a graph obtained by translating the points on the graph of y = x2 , a parabola, by the shift (h,k) and then applying the parameter a as a positive or negative vertical scaling, positive or negative depending on the sign of a, to the (h,k) translated or shifted parabola y = x2 The translated and vertically rescaled parabola has an axis of symmetry x = h.
If a is positive, the graph of parabola and its axis of symmetry is pitchfork that opens up ward. If a is negative, the graph of the parabola and its axis of symmetry is a pitchfork opening downward. B. Completing the Square, Effect of By completing the square, each quadratic ax2+bx+c = a[(x-h)2 + k ] with h = -b/(2a) and k = (4ac-b2)/(4a2). The graph of y = ax2+bx+c = a[(x-h)2 + k ] has an axis of symmetry with equation
C. Axes of Symmetry and Zeroes. If the a > 0, the quadratic has a minimum on the axis of symmetry, around which the parabola opens upward. The graph of y = ax2+bx+c = a[(x-h)2 + k ] along with the axis of symmetry then looks like a pitchfork opening upwards. If the a < 0, the quadratic has a maximum on the axis of symmetry, around which the parabola opens downward. The graph of y = ax2+bx+c = a[(x-h)2 + k ] along with the axis of symmetry then looks like a pitchfork opening upwards.
If k < 0 then solutions of the quadratic equation ax2+bx+c = 0 are also given by
If you are given that or show that ax2+bx+c = a(x +s)(x+r) then x = -s and x = -r give one or two x-intercepts of y = ax2+bx+c, and the axis of symmetry is at x = -½(r+s) = -b/(2a), halfway between the two intercepts. You may show that show that ax2+bx+c = a(x +s)(x+r) with factoring by inspection (if it works) or via two steps: (i) completing the square and (ii) factoring via difference of two squares if a difference of two squares results from the completing the square. |
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www.whyslopes.com
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