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 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.
8. Quadratics: Backward and Foreword use of Different Formulas
To draw a building which occupies a rectangle with the aid of coordinates, we would choose or take coordinate axes parallel to the sides of the rectangle as line segments parallel to coordinate axes are easily described using equations of the form x = a and y =b. Then to obtain small numbers a and b in those equations, we would place the origin of the coordinate system at one corner of the rectangle so that two sides lie on the coordinate axes. Alternatively, if symmetry might be advantages in our future reasoning or calculations, the origin might be placed at the center of the rectangle or offset, on one axis of symmetry, not both. So in choosing a coordinate system, we have options that may be taken to aid future work, or the determination of equations. Geometrical consideration of what coordinate system to use or which equation to use as the first step in further work are useful not only in drawing, but also but also in mathematics and physics, and indeed in any situation where coefficients in equations have to be found, etc, etc.
The equivalent expression
y = ax2+bx+c and y = a[(x-h)2+k] and y = a(x-h)2+ q
when obtained from each other through algebra, that is, through repeated use of the distributive law and/or factorization, all give the same value for y. In a given problem or situation, each expression has or may have an ease of use advantage over the other. Experience is needed to recognize the advantages if any.
Forward or Direct Use: One or more of the above expressions may be used to graph a quadratic or to draw conclusions about it. That say represents the forward or direct use of these expressions.
Backward Problem: Given data about the graph of a quadratic, find a formula for it. The or a solution is to to find the coefficients in one of the following equivalent equations for a quadratic.
y = ax2+bx+c and y = a[(x-h)2+k] and y = a(x-h)2+ q
So from the data, you may want to find the coefficients a, b and c (option I), or find the coefficients a, h and k (Option II), or find the coefficients a, h and q (Option III).
Which option requires the least amount of effort or work? That question itself, requires familiarity (experience) with graphing quadratics using these three equivalent expressions for them, and in graphing to understand the geometric & numeric meaning or use or significance of the coefficients in option I, II and/or III.
After the next example, see the remark about extensions of this problem in which the solution requires you to find a formula for the quadratic, and then draw some conclusions from the formula. That is a backward find the equation in some form followed by use the equation as is or in another form.
Example. Find an equation for the following parabolic dish.
given the low point has height 1 unit above the horizontal axis in the
illustration. .
Solution: We may use one of three (or four) forms for the equation of a parabola.
(A) y = ax2+bx+c
(B) y = a[(x-h)2+k] and
(C) y = a(x-h)2+ q
Which one is best remains to be determined below.
We add an axis of symmetry to the above diagram.
The parabola has an axis of symmetry x = h = ½(10+0) = 5. That gives the parameter h in two of the three equations..
Now the parabola has an extrema y = 1 on the axis of symmetry x = 5. So equation (C) yields y = 1 = q. So now we have two of the three unknown parameters in equation (C). That leaves one paramter a in equation (C) to determine. Equation (C) now says y = a(x-h)2+ q or
y = a(x-5)2+ 1
Now we use that y = 6 when x = 0 and when x = 10. Here x = 10 in the last equation forces, gives or implies that
6 = a(10-5)2+ 1 or equivalently
6 = a(5)2+ 1 or 6-1 = 25a or 5 = 25a or a = 1/5
Thus a = 1/5. So equation (C) becomes
y = a(x-5)2+ 1 = 1
5[(x-5)2+ 1]
So the equation of the parabolic dish expressed in standard form is
y = 1
5[(x-5)2+ 1]
Backward and Forward Problem Combined: Solve the backward problem and the answer a question about the quadratic using the result of the backward problem.
Remark: There is a forth option, a variant of the second perhaps. If quadratic y = ax2+bx+c has roots (zeroes) when x = u and x = v then another equivalent expression for it is y = a(x-u)(x-v). This fourth option is available only when there are roots (zeroes), that is when and only when the graph of the quadratic touches or crosses the x-axis. Then in the equation for axis of symmetry x = h, the parameter h = ½ (u+v).
Remark: Having so many options for the determination of a quadratic from geometric data may be confusing, but experience will or would help provide familarity with the options and so build your algebraic skills and concepts. The forward and backward use of quadratics is a frequent part of calculus and beyond that mathematics in physics and engineering,
www.whyslopes.com
Analytic Geometry
& Functions, etc
Area Entrance
Entrance + Pages Below this page
Pages at Current LevelGraphing Exercises Graph y = a[(x-h)^2 +k] Theory Factoring Quadratics Difference of Two Squares Completing the Square Convert to Standard Form (Arith) Quadratic Formula Finding Coefficients Applications Quadratics Summary Exercises
Area Entrance (C) Complex Numbers (FN) What are Functions? (FN) Functions - More SZM: Sign Analysis (L) Lines Summary (P) Polynomials (*,+,-) (Q) Quadratics (D) Simplify Square Roots (T) Unit Circle Trig Conic Sections Links More Links
Pages at Above this Page
Extras: Not all perfect.
Equal Sign Use/Abuse Real Numbers Say More Positive Linear Inequalities Triangle Inequality Absolute Value |x| |x| Eq'ns & Inequalities Rectangular Coords Shortest Path Distance Formulas Add & Multiply Points Polar Coordinates Radians (A) Vectors (A) Coordinate Arithmetic (A) Navigation on Maps (A) Addition Geometrically (A) Rotation (PT) Translations (PT) Dilatations PT: Rotations
Links to Site Pages outside this site area follow - co- and pre- requisites.
Easy Consequences of this (newest) Complex Number. Starter Lesson follow below to provide an alternate development of HS or college maths.
Vec & Cmplx No Applet
B2 C. Conjugates
B3 Pythagoras
B4 Distance
B5 Rt Triangle Similarity
B6 Trig., Functions
B7 Dot & Cross Products
B8 Cosine Law
B9 Exponential & cis fns
B10 Easy Trig Identities
B11 Set Viewpoint Arithmetic Videos - Real Player Format
Decimal Addition Methods
Decimal Subtraction Methods
Decimal Multiplication Methods
Decimal Division Methods
Fractions
Primes
Greatest Common Divisor
Divisors
Least Common Multiples
Square Root
SimplificationUsing formulas forwards & Backwards - A unifying theme for algebra skill development - the 4th skill in Volume 2, Three Skills for Algebra!