8. Quadratics: Backward and Forward use of 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
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y = a(x-5)2+ 1 =
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1
5
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[(x-5)2+ 1]
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So the equation of the parabolic dish expressed in standard form is
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,
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Unsolicited Advice
Learning to do and high marks if it comes to easy is often
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Appetite.
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