Find a polar equation of the form r=f(θ) for the curve represented by the cartesian equation x=−y2.

The polar equation of the curve .

Step-by-step explanation:

Given: Polar equation of the form for the curve represented by the cartesian equation .

Polar coordinates are defined in complex number as:

So, substituting the value in given equation:

                               

Hence, the polar equation of the curve of the form represented by the cartesian equation is  .

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We define the following variables:
 x = r * cos (θ)
 y = r * sine (θ)
 Substituting the variables we have:
 x = -y ^ 2
 r * cos (θ) = - (r * sin (θ)) ^ 2
 Rewriting:
 r * cos (θ) = - (r ^ 2 * sin ^ 2 (θ))
 We cleared r:
 r = - ((cos (θ)) / (sin ^ 2 (θ)))
 We rewrite:
 r = - ((cos (θ)) / (sin (θ))) * (1 / sin (θ))
 r = - cot (θ) * csc (θ)
 
 a polar equation of the form r = f (θ) for the curve represented by the cartesian equation x = -y2 is:
 r = - cot (θ) * csc (θ)

The axis of symmetry is x=5

Step-by-step explanation:

Given function is :

f(x)=

We can use the formula for the axis of symmetry.

Therefore, the axis of symmetry is x=5

Axis of symmetry is the line of symmetry of a parabola that divides it into two equal halves. These halves are reflections of each other about the line of symmetry.