# Thread: find intercepts, test for symmetry

1. ## find intercepts, test for symmetry

a cardiod microphone pattern is given by the following equation:

$\displaystyle \left{({x}^{2}+{y}^{2}-x\right)}^{2}={x}^{2}+{y}^{2}$

the expanded form of this equation is
$\displaystyle x^4-2 x^3+2 x^2 y^2+x^2-2 x y^2+y^4 = x^2+y^2$

(a) Find the intercepts of the graph of the equation

to find $\displaystyle {x}_{intercepts}$ I set $\displaystyle y = 0$

thus resulting in

$\displaystyle x^4-2x^3=0\Rightarrow x^3\left( x-2 \right)$

so $\displaystyle {x}_{intercepts} = 0, 2$

to find $\displaystyle {y}_{intercepts}$ I set $\displaystyle x = 0$

thus resulting in

$\displaystyle Y^2=1$ so $\displaystyle {y}_{intercepts}$ are $\displaystyle \pm 1$

(b) Test for symmetry with respect to the x-axis

my questions is that i do not how the symmetry is found except by observation from the graph. so symmetry is only on x-axis not on y-axis.

the books answers were (a)$\displaystyle \left( 0,0 \right),\left( 2,0 \right) \left( 0,1\right) \left(0,-1 \right)$(b)x=axis symmetry

but wolfframalpha gave something else

btw like the upgrades to latex

2. Originally Posted by bigwave
a cardiod microphone pattern is given by the following equation:

$\displaystyle \left{({x}^{2}+{y}^{2}-x\right)}^{2}={x}^{2}+{y}^{2}$

the expanded form of this equation is
$\displaystyle x^4-2 x^3+2 x^2 y^2+x^2-2 x y^2+y^4 = x^2+y^2$

(a) Find the intercepts of the graph of the equation

to find $\displaystyle {x}_{intercepts}$ I set $\displaystyle y = 0$

thus resulting in

$\displaystyle x^4-2x^3=0\Rightarrow x^3\left( x-2 \right)$

so $\displaystyle {x}_{intercepts} = 0, 2$

to find $\displaystyle {y}_{intercepts}$ I set $\displaystyle x = 0$

thus resulting in

$\displaystyle Y^2=1$ so $\displaystyle {y}_{intercepts}$ are $\displaystyle \pm 1$

(b) Test for symmetry with respect to the x-axis

my questions is that i do not how the symmetry is found except by observation from the graph. so symmetry is only on x-axis not on y-axis.

the books answers were (a)$\displaystyle \left( 0,0 \right),\left( 2,0 \right) \left( 0,1\right) \left(0,-1 \right)$(b)x=axis symmetry

but wolfframalpha gave something else

btw like the upgrades to latex
There are three typical symmetries looked at...Reflection over the x axis, reflection over the y axis and reflection over the line y = x.

If the substitution y --> -y gives the same equation, then the equation is symmetric over the x axis.

If the substitution x --> -x gives the same equation, then the equation is symmetric over the y axis.

If the substituions x --> y and y --> x (ie switch x and y in the equation) gives the same equation, then the equation is symmetric over the x axis.

This is a long-winded way to say that I agree with your answer for b).

-Dan

3. In order to test for symmetry, check these conditions:

If f(x) = f(-x), then the graph is symmetric on the Y-axis.

If f(y) = f(-y), then the graph is symmetric on the X-axis.

Does this make sense why these two conditions give you symmetry?

4. this was a little confusing because it appears to be an implicit equation. so doesn't mean we really don't have a function since it fails the vertical line test?

5. Originally Posted by bigwave
this was a little confusing because it appears to be an implicit equation. so doesn't mean we really don't have a function since it fails the vertical line test?
You are correct...your equation is not a function. However the symmetry tests still work.

-Dan

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