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YOU are better than YOU think. Show
yourself how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work
and study.
Read notes and textbooks like a lawyer, so that no nuance, no subtlety and no clause escapes your attention. |
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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
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
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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Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
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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.
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck.
| |
Discussion of the standard form y = a[(x-h)2+k]
Dependence on Parameters, Extrema and Zeros
from standard form. Special Case of the Quadratic formula for the standard
form.
The standard-form-for-graphing
y = a[(x-h)2+k]
represents two the result of two operations on the curve y = x2, a
parabola that opens up with an axis of symmetry x = 0. The case a = 0 would give
y = 0. So we suppose a is non-zero.
The first operation translates the points on y = x2. by adding (h,k)
to them. That moves the axis of symmetry to x = h and raises or lowers the
height of each point by |k| = the magnitude of k.
The graph of y = (x-h)2+k is parabola with minimum at x =
h and an axis of symmetry x = h. This parabola and its axis of symmetry
together look like an upward pointing pitchfork
If k > 0, the curve y = x2 is moved upward, and the equation
0 = y = (x-h)2+k
has no solutions. So there are no x-intercepts.
If k = 0, the curve is not moved upward, and the equation
0 = y = (x-h)2+k = (x-h)2
has one solution, given by x - h = 0. So there is only one
x-intercept, namely x = h.
Finally, if k < 0, then the equation
0 = y = (x-h)2+k
has solutions, given by the values of x which satisfy
(x-h)2 = -k > 0
which are provided by the two equations x - h = (-k)½
and x - k = (-k)½. Hence x = h plus or minus (-k)½
gives two solutions of
0 = y = (x-h)2+k
and hence the location of two x-intercepts.
In all three cases, the graph of y = (x-h)2+k is parabola
with minimum at x = h and an axis of symmetry x = h. This parabola and its
axis of symmetry together look like an upward pointing pitchfork.
The first operation translates the points on y = x2. by adding (h,k)
to them. That moves the axis of symmetry to x = h and raises or lowers the
height of each point by |k| = the magnitude of k. The value of k
determines whether or not the parabola part crosses the x-axes, twice, once or
nonce times.
The second operation follows the translation by a vertical scaling of the
y-values or coordinates, that is a vertical multiplication by the nonzero scale
factor a of the points on the upward direct pitchfork y = (x-h)2+k
to obtain the graph of y = a[ (x-h)2+k.].
The inclusion of the nonzero factor a implies 0 = a[ (x-h)2+k.].when
and only when 0 = [ (x-h)2+k]. So the previous
analysis gives the number and/or locations of the zeroes or y-intercepts.
The number and/or locations depend only the values of h and k.
Now if a > 0, the second operation results in an parabola that opens up
with a axis of symmetry x = h. The case a > 1 implies the parabola
height increases more rapidly than that of the parabola y = (x-h)2+k.
The case 0 < a < 1 implies the parabola height increases less rapidly than
that of the parabola y = (x-h)2+k
Now if a < 0, the second operation results is equivalent to multiply first
by |a| and then multiplying by -1. After both multiplications, the
case |a| > 1 implies the parabola height decreases more rapidly than that of
the parabola y = (x-h)2+k. The case 0 < |a| < 1 implies the
parabola height decreases less rapidly than that of the parabola y = (x-h)2+k.
The axes of symmetry together the graph of y = a[ (x-h)2+k.]
looks like a pitchfork that points and opens downward.
Location of Zeroes
If k < 0, then y = (x-h)2 + k = 0 when and only when
(x-h)2 = -k or
| x-h =± |
__
Ö-k |
or |
x = h ± |
__
Ö-k |
This gives the first way to solve a[(x-h)2 + k ] = 0 or ax2+bx+c
= 0 when ax2+bx+c = a[(x-h)2 + k ]. The solutions are
equidistant from the axis of symmetry, the line x = h.
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Graph y = a[(x-h)^2 +k] Theory
Factoring Quadratics
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Convert to Standard Form (Arith)
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Skills for Algebra!
What
is a Variable?
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