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Equation 1:  r = 4 +( 3 * cos theta  )

Equation 2: r = sqrt ( 5² * sin(2 theta) )

Step-by-step explanation:

GRAPH 1:  

The first graph is a dimpled limacon.

General equation for dimpled limacon:

r = a + b cos theta                     ∴ if dimple is along the x- axis

r = a + b sin theta                      ∴ if dimple is along the y-axis

y-intercept : { a, -a }  = { 4, -4 }   ∴ the points at which limacon intersects y-axis

Negative side of x-axis = ( a – b ) ⇒ 1

Positive side of x-axis = ( a + b ) ⇒ 7

Subtract the value of a from sum of a and b to find b:

b = 7 – 4 ⇒ 3

Equation1:  r = 4 +( 3 * cos theta  )

GRAPH 2:  

The second graph is a lemniscates.  

General equation for lemniscates is:

r² = a² cos(2theta)                        ∴ if petals of graph are on coordinate axis

r² = a² sin(2 theta)                ∴ if petals of graph are not on coordinate axis

now, according to the graph:

a = 5 ⇒ a² = 25

angle of graph:  cos2θ, simply divide 360° by 2:

⇒ 180°

The petals cannot be on coordinate axis, we start from 45° and then the next petal will be on:

45° + 180° = 225°

Since the graph is not on the coordinate axis, so

r² = 5² sin(2 theta)   ⇒    r = sqrt ( 5² * sin(2 theta) )      

Equation 2: r = sqrt ( 5² * sin(2 theta) )      

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step-by-step explanation: