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,