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Worked solution

Check exact intercepts of four rational curves

Find the exact coordinates of the axial intercepts for the following curves.

(a) y=1/(3−x)y=1/(3-x).

(b) y=(2x+1)/(x−2)y=(2x+1)/(x-2).

(c) y=x2/(x−1)y=x^2/(x-1).

(d) y=(x2+2x−3)/(x+2)y=(x^2+2x-3)/(x+2).

CJC · Lecture · Self-Practice 1

(a) Nonzero numerator

The source gives y-intercept (0,1/3)(0,1/3) and no x-intercept. Independently, x≠3x\ne3; substituting x=0x=0 gives 1/31/3, and the numerator 11 never vanishes.

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(b) Linear numerator

The source gives y-intercept (0,−1/2)(0,-1/2) and x-intercept (−1/2,0)(-1/2,0). Independently, x≠2x\ne2; solve 2x+1=02x+1=0 and check its denominator is nonzero.

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(c) The origin lies on both axes

The source gives (0,0)(0,0) as both intercepts. Independently, x≠1x\ne1; the numerator vanishes only at 00, where the denominator is −1-1.

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(d) Factor before solving

The source gives y-intercept (0,−3/2)(0,-3/2) and x-intercepts (−3,0),(1,0)(-3,0),(1,0). Independently, x≠−2x\ne-2 and x2+2x−3=(x+3)(x−1)x^2+2x-3=(x+3)(x-1). Neither root is excluded; substituting zero gives −3/2-3/2.

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Common mistakes

A zero of the numerator is an intercept only when the original denominator is nonzero; one origin point may be both axial intercepts.

Solution notes

Both Student and Teacher originals compared. Printed answers retained; independent explanations identified in the working. No printed marks or external attribution.

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