document.write( "Question 1116164: Graph r(x)=(x-3)^2/x^2-4 \n" ); document.write( "

Algebra.Com's Answer #730996 by greenestamps(13200) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "I will guess that you mean $\"%28x-3%29%5E2%2F%28x%5E2-4%29\"$ instead of $\"%28x-3%29%5E2%2Fx%5E2-4\"$ , because the first is far more interesting than the second.

\n" ); document.write( " $\"%28x-3%29%5E2%2F%28x%5E2-4%29+=+%28x-3%29%5E2%2F%28%28x%2B2%29%28x-2%29%29\"$

\n" ); document.write( "The function has a double root at x=3, where the numerator is zero, and vertical asymptotes at x=-2 and x=2, where the denominator is zero.

\n" ); document.write( "To get an idea of where the function is positive or negative, imagine \"walking\" along the x axis left to right, starting at a large negative value of x.

\n" ); document.write( "The function consists of 4 linear factors; for large negative values of x, all 4 factors are negative, so the function is positive.

\n" ); document.write( "When you pass the vertical asymptote at x = -2, the sign of one of the 4 factors changes, so the function value changes from positive to negative.

\n" ); document.write( "At x=2 you pass another vertical asymptote; one factor changes sign; so the function value changes sign again, from negative to positive.

\n" ); document.write( "When you pass the zero at x=3, TWO of the 4 factors change sign at the same time, so the function does NOT change sign. The graph touches the x axis but stays positive.

\n" ); document.write( "Finally, for the end behavior, observe that for large negative or large positive values of x the function is very close to $\"x%5E2%2Fx%5E2+=+1\"$ ; so the graph has a horizontal asymptote at y=1.

\n" ); document.write( "Here is a graph....

\n" ); document.write( " $\"graph%28400%2C400%2C-5%2C5%2C-10%2C10%2C%28x-3%29%5E2%2F%28x%5E2-4%29%29\"$ \n" ); document.write( "

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