Microphones In studios and on stages, cardioid microphones are often preferred for the richness they add to voices and for their ability to reduce the level of sound from the sides and rear of the microphone. Suppose one such cardiod pattern is given by the equation (x2 + y2x)2 = x2 + y2.

  • (a) Find the intercepts of the graph of the equation.
  • (b) Test for symmetry with respect to the x-axis, y-axis, and origin.
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(a) Intercepts: When y=0, (x^2-x)^2=x^2; x^2-x=+x. So we have x^2-x-x=0 or x^2-x+x=0. x(x-2)=0 or x^2=0. So x=0 or 2; when x=0, y^4=y^2 and y^2(y-1)(y+1)=0, so y=0, 1 or -1. Therefore we have intercepts at (0,0), (0,1), (0,-1) and (2,0). That is, three on the y axis and 2 on the x axis (the origin is an intercept for x and y). The origin is the cusp of the cardioid and the body of the cardioid cuts the x axis at x=2.

(b) If we put -y into the equation we get the same results for x as we do when we put +y into the equation. So the graph is symmetrical about the x axis. If we put -x into the equation, the contents of the brackets becomes (x^2+y^2+x), which is not equal to (x^2+y^2-x) when we have +x. So even though this expression is squared, it's the square of a different number. So the graph is not symmetrical about the y axis. To check this out, put x=1 in the equation: y^4=y^2+1, y^4-y^2-1=0, so y^2=(1+sqrt(5))/2 and y is plus or minus the right hand side (above and below the x axis); now put x=-1: (y^2+2)^2=y^2+1; y^4+3y^2+3=0, which has no real roots, so the graph curve doesn't exist at x=-1, which is to the left of the origin.

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