Graphs and Solutions Test Answer Key with Scoring – Algebra 1

Question 1 Points on a Parabola (2 points)

(2, −3): 4 − 4 − 3 = −3 ✓. (−1, 2): 1 + 2 − 3 = 0, not 2. (4, 5): 16 − 8 − 3 = 5 ✓.

(2, −3) and (4, 5)

Scoring: 1 point for checking all three points, 1 point for the correct choice.

Question 2 Line and Parabola (3 points)

1. x² − 4x + 3 = x − 1, or x² − 5x + 4 = 0.
2. (x − 1)(x − 4) = 0, so x = 1 or x = 4.
3. y = x − 1 gives 0 and 3: (1, 0) and (4, 3).

Scoring: 1 point per part.

Question 3 Estimate Then Confirm (3 points)

1. The graphs cross near (0, −3) and (3, 0).
2. x² − 2x − 3 = x − 3 gives x² − 3x = 0, so x(x − 3) = 0 and x = 0 or x = 3. Then y = x − 3 gives −3 and 0: (0, −3) and (3, 0).

Scoring: (a) 1 point. (b) 1 point for the equation and factoring, 1 point for both points.

Question 4 No Intersection (2 points)

x² + 1 = 2x − 3, so x² − 2x + 4 = 0. The discriminant is 4 − 16 = −12, which is negative.

There is no real solution, so the graphs never meet.

Scoring: 1 point for the equation, 1 point for the discriminant and conclusion.

Question 5 Exponential Graph (3 points)

1. The graph crosses the y-axis at (0, 5) and the values halve each step: y = 5(12)^x.
2. As x → ∞, y → 0; as x → −∞, y → ∞.
3. 5(12)⁻⁵ = 5 · 32 = 160.

Scoring: 1 point per part.

Question 6 Monthly Growth Factor (3 points)

1. A = 800(1.12^1/12)^12t ≈ 800(1.0095)^12t. The monthly factor is about 1.0095, a rate of about 0.95%.
2. t = 0.5: 800(1.12)^0.5 = 800(1.0583) ≈ $846.64.

Scoring: (a) 1 point for the power rule, 1 point for the factor. (b) 1 point.

Question 7 System Solution Check (2 points)

Check (3, 9): 3² = 9 ✓ and 4(3) − 3 = 9 ✓. The other solution comes from x² = 4x − 3, so x² − 4x + 3 = 0 and (x − 1)(x − 3) = 0, giving x = 1 and y = 1.

The other solution is (1, 1).

Scoring: 1 point for the check, 1 point for the other solution.

Question 8 Rewrite an Exponent (2 points)

5^{x + 2} = 5^x · 5² = 25 · 5^x.

At x = 1: 25 · 5 = 125. Check: 5³ = 125.

Scoring: 1 point for the form 25 · 5^x, 1 point for the value.