Rational Functions Test Answer Key with Scoring – Algebra 2

Question 1 Domain and asymptotes (2 points)

The denominator is 0 at x = −6. Equal degrees give y = 4/1 = 4.

Domain: all reals except −6; vertical asymptote x = −6; horizontal asymptote y = 4

Scoring: 1 point for the domain and vertical asymptote, 1 point for the horizontal asymptote.

Question 2 Transformations (2 points)

The factor −7 stretches vertically by 7 and reflects across the x-axis; x − 5 shifts right 5; −2 shifts down 2.

Reflect, stretch by 7, right 5, down 2

Scoring: 1 point for the reflection and stretch, 1 point for the two shifts.

Question 3 From graph to equation (3 points)

a) The dashed lines are x = −1 and y = −2.

b) y = ax + 1 − 2. At (0, 2): 2 = a − 2, so a = 4. Check (3, −1): 4/4 − 2 = −1 ✓.

y = 4x + 1 − 2

Scoring: 1 point for the asymptotes, 1 for a, 1 for the equation.

Question 4 Hole, asymptotes (3 points)

Factor: (x − 5)(x + 4)(x − 5)(x + 5) = x + 4x + 5 for x ≠ 5.

Hole at x = 5 with height 9/10. Vertical asymptote x = −5. Horizontal asymptote y = 1.

Hole (5, 910); x = −5; y = 1

Scoring: 1 point each feature.

Question 5 Inverse variation (3 points)

a) k = 7 · 8 = 56, so y = 56x.

b) y = 56/14 = 4.

c) 2 = 56/x, so x = 28.

Scoring: 1 point per part.

Question 6 Average cost (3 points)

a) A(150) = 5 + 900/150 = 5 + 6 = $11.

b) 5 + 900/n = 8, so 900/n = 3 and n = 300.

c) A = 5: as more items are printed, the average cost approaches the $5 cost per item.

Scoring: 1 point per part.

Question 7 Rewrite and identify (2 points)

4x − 7 = 4(x − 3) + 5, so 4x − 7x − 3 = 4 + 5x − 3.

y = 5x − 3 + 4; asymptotes x = 3 and y = 4

Scoring: 1 point for the rewrite, 1 point for the asymptotes.

Question 8 A hole on a graph (2 points)

a) (x − 5)(x + 5)x + 5 = x − 5, so the hole is at x = −5 with height −10: (−5, −10).

b) All real numbers except −5.

Scoring: 1 point per part.