Question

Which graph represents the rational function ?

Answer 1

I don't see the function or the candidate graphs. Could you paste the rational function and the graphs (or upload an image)?

While you do that, here is a short checklist you can use to match any rational function to a graph:

1. Simplify and factor numerator and denominator.
- If a factor cancels, there is a hole at that x-value (not a vertical asymptote).

2. Vertical asymptotes:
- Zeros of the denominator that do not cancel are vertical asymptotes. Note behavior near each asymptote (±∞ on left/right).

3. Holes:
- For a cancelled factor (x−a), the graph has a hole at x = a; find y by plugging x = a into the simplified form.

4. x-intercepts:
- Zeros of the numerator that are not cancelled are x-intercepts. Multiplicity matters:
- Odd multiplicity: crosses the axis.
- Even multiplicity: touches and turns.

5. y-intercept:
- Evaluate f(0) (unless x=0 is a hole).

6. End behavior / horizontal or oblique asymptote:
- deg(numerator) < deg(denominator): horizontal asymptote y = 0.
- deg(numerator) = deg(denominator): horizontal asymptote y = ratio of leading coefficients.
- deg(numerator) = deg(denominator) + 1: slant (oblique) asymptote found by polynomial long division.
- higher difference: polynomial asymptote from division.

7. Sign chart / sample points:
- Between vertical asymptotes and beyond, pick test x-values to determine whether the curve is above or below the asymptotes.

If you post the function and the available graphs (or describe them), I’ll identify which graph matches and show the reasoning.