Discussion of standard form \[y = a[(x-h)^2+k]\]
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
\[ x - h =\pm\sqrt{k} \mbox{ or } x - h = -\sqrt{k}\]
Hence \[x = h \pm\sqrt{k} \] 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. That is when
\begin{eqnarray*} x - h =\pm\sqrt{k} & \mbox{ or } x = h \pm\sqrt{k}
\end{eqnarray*}
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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