The principles/ideas that help in visualizing the curve or rough sketching the curve are:

**1. Origin and tangents at the origin**

Check whether the curve passes through the origin and finding out whether (0,0) satisfies the curve equation.

If the curve passes through (0,0) find the equation of tangent at (0,0)

**2. Points of intersection of the curve with the coordinate axes**

You can find the point at which the curve crosses the x-axis by putting y = 0 in the curve equation.

Similarly putting x = 0 gives the point at which the curve passes y-axis.

**3. Symmetry**

If all powers of y in the curve equation are even, then the curve is symmetric about x-axis.

The familiar curve y² = 4ax is symmetric about x-axis.

If all powers x in the equation of curve are even, it is symmetric about y-axis.

If by putting x = -x and y = -y, the equation of the curve remains the same, then it symmetric in opposite directions.

If the equation remains unaltered by interchanging x and y in the equation, then it is symmetric about the line y =x.

**4. Regions where the curve does not exist**

In the regions where the curve does not exist, x or y values will be imaginary.

In the case of y² = 4ax, when x is negative, y values will be imaginary. Hence curve does not exist on the left side of y-axis.

**5. Find points where dy/dx = 0**

At these points, the slope of the tangent to the curve is zero and the tangent will be parallel to the x axis.

**6. Find maxima and minima points**

Find points where dy/dx = 0 and check the sign of d²y/dx² at these points and find out maximum (d²y/dx² is negative) and minimum (d²y/dx² is positive) points of the curve.

Also find the interval in which the dy/dx is g.t. 0. In this interval, the function is monotonically increasing.

In the interval in which dy/dx is l.t. 0, the function is monotonically decreasing.

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