# $y^{2}$=$(x-a)(x-b)^{2};a,b>0,a>b,$

Toolbox:
• The strategies for curve tracing
• (1) Domain-The values of x for which the function is defined
• (2)Extent -Horizontal (vertical ) extent is determined by the intervals for $x(y)$ for which the curve exists
• (3) $x=0$ gives the y intercept and y=0 gives the x -intercept
• (4) If $(0,0)$ satisfies the equation, the curve passes through the orgin
• Symmetry:
• (5) Find out whether the curve is symmetrical about any line. The curve is symmetric about
• (i) x-axis if the equation is unaltered when y is replaced by -y
• (ii) y-axis if the equation is unaltered when x is replaced by -x
• (iii)the orgin if the equation is unattened when x is replaced by -x abd y by -y
• (iv) the line $y=x$ if the equation is uttered when x and y are interchanged
• (v) the line $y=-x$ if the eauation is unchanged if x and y are replaced by -y and -x
• (6) Asymptotes ($\parallel$ to coordinate axes)
• If $y \to c$ (finite) when $x \to \pm \infty$
• (or $x \to k$ (finite ), when $y \to \pm \infty$
• the line $y=c$(or $x=k$ ) is an asymptole parallel to x-axis (or y-axis)
• (7)Monotsnicity : Determine the intervals for which the curve is increasing or decreasing, using the first derivative test
• (8) Special points: Determine the intervals of concavity/convexity and the points of inflection, using the second derivative test.
$y^2=(x-a)(x-b)^2\;a ,b > 0, a >b$
Step 1:
Existence : The curve does not exist for $x < a$
$\therefore$ it does not exist for $-\infty < x < a$ except at $x=b( b < a)$ when $y=0$
The curve exists for $a \leq x < \infty$
Step 2:
(ii) Symmetry: The curve is symmetric about x-axis only
Step 3:
(iii) Asymptotes: The curve does not admit asymptotes since there are no further values of $x(or\;y)$ for which $y \to \pm \infty(or\; x \to \pm \infty)$
Step 4:
Loops:The curve passes through (0,0) twice but no loop is formed because the curve does not exist for $b \leq x < a$