graphs and transformations of quadratic equations

1. Graphs of 𝒚𝟐
= 𝒙 and 𝒚𝟐
= −𝒙
Note that the graph of 𝑦2
= 𝑥 is obtained just by interchanging x with
y and y with x in the equation 𝑦 = 𝑥2
. Also The graphs of 𝑦2
= 𝑥 and
𝑦2
= −𝑥 are given hereunder for understanding and reference. The
functions are not single valued; for a single value of x there are two
values of y. We would still study such double valued functions as their
graphs are as important as graphs of single valued functions.. But we
call relations 𝑦 = 𝑓(𝑥) = 𝑥2
and 𝑦 = 𝑔(𝑥) = −𝑥2
as single valued
functions or simply functions. In such situation, it may not be advised
that the variables x and y may be interchanged, problem worked out
and then x and y interchanged again. It may work in some cases. But
that would be wrong in general, for, many properties of the graph are
changed if we shift the axes, or rotate of flip them. If we want to
rigorously stick to the definition of function as single valued these
two kinds of equations are strictly not functions. There may be many
values of x for same value of y but not vice versa.
It may be repeated with emphasis, that if the graph is changed in any
way, the equation is changed, in consequences, the roots may be
changed. For example, if one shifts the old x-axis to the vertex of the
graph, the two different roots are changed and the new roots become
equal. The equation is albeit changed (see for yourself).
One may think that the graph of the equation 𝑦2
= 𝑥 is obtained by
interchanging x for y and y for x in the equation 𝑥2
= 𝑦 and that
results in a different equation. This is always not the case. For
example, the equation y = x does not change even if we interchange

2. the variables.
We see that these graphs are also parabolas, with their axis of
symmetry being along the x-axis instead of along the y-axis.
We have seen that the graph of 𝑦 = 𝑎𝑥2
, 𝑦 = 𝑎𝑥2
+ 𝑏𝑥 + 𝑐 etc. are
got by scaling the graph of 𝑦 = 𝑥2
and scaling and shifting the graph
of 𝑦 = 𝑥2
without changing their axis of symmetry.
Parabolas whose axis of symmetry is not parallel to x-axis or y-axis
Can we have parabolas having their axis of symmetry in any
direction, not particularly along the x-axis or y-axis? If we can have
such parabolas, is their corresponding equation a quadratic
equation? If they are, are they of the type 𝑦 = 𝑎𝑥2
+ 𝑏𝑥 + 𝑐 or any
other expression not like this? The key question is, if we rotate a
parabola through an angle, what is the result?
Graph of 𝑥 = 𝑦2
Graph of 𝑥 = −𝑦2

3. Let there be a parabola 𝑌 = 𝑋2
in XY axes (Black Axex) and
B(X,Y) be any point on it, described by 𝑋 = 𝑡, 𝑌 = 𝑡2
, describing
the equation 𝑌 = 𝑋2
in terms of a single parameter ‘t’ instead of two
variables ‘X’ and ‘Y’, while keeping relationship between the two
variables intact. Such an equation like 𝑋 = 𝑡, 𝑌 = 𝑡2
is called
‘parametric equation’ of the parabola 𝑌 = 𝑋2
. There can be many
parametric forms as such. The XY axes are at an angle θ with our
standard xy-axes (Blue Axes). From B, drop a perpendicular BA on
the X-axis, and a perpendicular BC on Y axis, and a perpendicular BR
on x-axis. From A drop a perpendicular AQ on the x-axis and from B
drop a perpendicular BP on QA extended. Let the ∠AOQ=θ. Since B
is the point 𝑋 = 𝑡, 𝑌 = 𝑡2
on the parabola in the XY axes, 𝑂𝐴 =
A
x’
x
O
C
B
y
y’
P
Q
R
θ
θ
t
𝑡2
X
Y

4. 𝑡, 𝐵𝐴 = 𝑡2
. It is easily seen that OABC is a rectangle, as well as
PBRQ. Apply similar triangles to ΔAOQ and ΔBOP, to show ∠BAP=θ.
We have to find out the coordinates of the point B ,i.e., OR and RB
and explore how they are related.
Now we can easily show that
𝑂𝑄 = 𝑡 cos 𝜃 , 𝐴𝑄 = 𝑡 sin 𝜃, 𝐴𝑃 = 𝑡2
cos 𝜃, and
𝐵𝑃 = 𝑡2
sin 𝜃.
So, −𝑂𝑅 = 𝑄𝑅 − 𝑄𝑂 = 𝐵𝑃 − 𝑂𝑄. (𝑂𝑅 = 𝑥).
Hence 𝑥 = 𝑡 cos 𝜃 − 𝑡2
sin 𝜃 …………………………...(1)
Similarly, 𝐵𝑅 = 𝑃𝑄 = 𝑃𝐴 + 𝐴𝑄
Or, 𝑦 = 𝑡 sin 𝜃 + 𝑡2
cos 𝜃, ………………………………..(2)
To find the relation between x and y we have to eliminate t from
these two equations.
Multiply eqn.(1) by cos 𝜃 and eqn.(2) by sin 𝜃 and add.
𝑥 cos 𝜃 = 𝑡 cos2
𝜃 − 𝑡2
sin 𝜃 cos 𝜃
𝑦 sin 𝜃 = 𝑡 sin2
𝜃 + 𝑡2
sin 𝜃 cos 𝜃
Adding, we get, 𝑥 cos 𝜃 + 𝑦 sin 𝜃 = 𝑡 cos2
𝜃 + 𝑡 sin2
𝜃
Or, 𝑥 cos 𝜃 + 𝑦 sin 𝜃 = 𝑡…………………………………..(3)
Putting this value of t in eqn. (2) we get,

5. 𝑦 = (𝑥 cos 𝜃 + 𝑦 sin 𝜃) sin 𝜃
+ (𝑥 cos 𝜃 + 𝑦 sin 𝜃)2
cos 𝜃
Or, 𝑦(1 − sin2
𝜃) − 𝑥 cos 𝜃 sin 𝜃 = (𝑥 cos 𝜃 +
𝑦 sin 𝜃)2
cos 𝜃
Or, 𝑦 cos2
𝜃 − 𝑥 cos 𝜃 sin 𝜃 = (𝑥 cos 𝜃 + 𝑦 sin 𝜃)2
cos 𝜃
Or, 𝑦 cos 𝜃 − 𝑥 sin 𝜃 = (𝑥 cos 𝜃 + 𝑦 sin 𝜃)2
………….(4)
This is the equation of the parabola 𝑌 = 𝑋2
in XY axes which
becomes 𝑦 cos 𝜃 − 𝑥 sin 𝜃 = (𝑥 cos 𝜃 + 𝑦 sin 𝜃)2
in the
regular xy-axes. Since we have just rotated the XY axes into the xy-
axes, the parabola remains the same but the equation changes.
A close look at equation (4) tells us that the XY-axis has rotated
through an angle – 𝜽 to become xy-axes and the equation 𝑌 = 𝑋2
has become 𝑦 cos 𝜃 − 𝑥 sin 𝜃 = (𝑥 cos 𝜃 + 𝑦 sin 𝜃)2
. In
other words, from equation (3), as 𝑡 = 𝑋, we may write
𝑥 cos 𝜃 + 𝑦 sin 𝜃 = 𝑋…………………………………..(5)
Again from eqn. (4), we may write,
𝑦 cos 𝜃 − 𝑥 sin 𝜃 = 𝑌…………………………………....(6)
The situation may be viewed from a different perspective. When XY-
axis has rotated through an angle – 𝜽 to become xy-axes, this is just
in other words, xy-axis has rotated through an angle 𝜽 to become
XY-axes. Now 𝐜𝐨𝐬(−𝜽)=𝐜𝐨𝐬 𝜽 and 𝐬𝐢𝐧(−𝜽)=− 𝐬𝐢𝐧 𝜽. So
we can rewrite eqn.(5) and eqn.(6) as
𝑋 cos 𝜃 − 𝑌 sin 𝜃 = 𝑥…………………………………..(5’) and

6. 𝑌 cos 𝜃 + 𝑋 sin 𝜃 = 𝑥…………………………………....(6’)
For now we have explored shifting, scaling and rotating the
coordinate axex, at least the orthogonal Cartesian axes. Note that
when the old axes are shifted by, h and k respectively (the origin at
(0,0) shifted to the point (h, k) to form a new set of axes, from the
perspective of new axes the old axes are shifted just by (−ℎ, −𝑘)
respectively. Similarly when any old axis is scaled (magnified) by a
factor ‘a’, from the perspective of new axes, the corresponding old
axis is scaled by a factor 1/a . Notice that shifting or rotation do not
change dimensions of the graph, although they change the equation
which the graph represents. Whereas scaling changes the dimension
of the graph as well as the equation of the graph. Flip about an axis is
rotation of the graph by 1800
.