Graphing Rational Functions — Quiz

Question 10 • Mathematics III

Which graph represents the function f of x is equal to the fraction with numerator 3 x minus 2 and denominator x minus 2 ?

Answer

Image with description A coordinate plane with a graph showing two branches. The branch on the left approaches y equals 2 and one-third as x approaches negative infinity from the right and approaches infinity as x approaches negative 2 from the left. Its turning point is near (negative 5 comma 5). The branch on the right approaches negative infinity as x approaches negative 2 from the right and approaches y equals 2 and one-third as x approaches infinity from the left. Its turning point is near (0 comma negative 1).

Image with description A coordinate plane with a graph showing two branches. The branch on the left approaches y equals 3 as x approaches negative infinity from the right and approaches negative infinity as x approaches 2 from the left. Its turning point is near (0 comma 1). The branch on the right approaches infinity as x approaches 2 from the right and approaches y equals 3 as x approaches infinity from the left. Its turning point is near (4 comma 5).

Image with description A coordinate plane with a graph showing two branches. The branch on the left approaches the x-axis as x approaches negative infinity from the right and approaches negative infinity as x approaches 2 from the left. Its turning point is near (1 comma negative 1). The branch on the right approaches infinity as x approaches 2 from the right and approaches the x-axis as x approaches infinity from the left. Its turning point is near (3 comma 1).

Image with description A coordinate plane with a graph showing two branches. The branch on the left approaches y equals one-third as x approaches negative infinity from the right and approaches infinity as x approaches 1 from the left. Its turning point is near (0 comma 1). The branch on the right approaches negative infinity as x approaches 1 from the right and approaches y equals one-third as x approaches infinity from the left. Its turning point is near (1 comma 0).

Explanation

Rewrite the function as (3x − 2)/(x − 2) = 3 + 4/(x − 2). This shows a vertical asymptote at x = 2 and a horizontal asymptote at y = 3. As x approaches 2 from the left, the function falls toward negative infinity; from the right, it rises toward positive infinity. Those features match choice B, which also has the branches approaching y = 3 at both ends.

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