Algebra I End-of-Course (EOC)

1. Question 1 · Solving Equations

   A climbing gym charges a one-time membership fee of $40 plus $25 per month. Jordan has paid a total of $340 so far. The equation 25m + 40 = 340 models the situation. How many months, m, has Jordan been a member?
   • A.12 months
   • B.15.2 months
   • C.13.6 months
   • D.8.5 months
   Show answer and explanation

   Answer: A. 12 months

   Subtract the fee first: 25m = 340 - 40 = 300, so m = 300 ÷ 25 = 12 months. Getting 15.2 comes from adding 40 instead of subtracting it, and 13.6 comes from ignoring the $40 fee and dividing 340 by 25.

2. Question 2 · Functions and Sequences

   The graph of a quadratic function f is a parabola with its vertex at (1, -4). The function g is defined by g(x) = f(x - 2) + 3. What is the vertex of the graph of g?
   • A.(-1, -1)
   • B.(3, -7)
   • C.(3, -1)
   • D.(4, -2)
   Show answer and explanation

   Answer: C. (3, -1)

   f(x - 2) shifts the graph 2 units right and +3 shifts it 3 units up, so the vertex moves from (1, -4) to (1 + 2, -4 + 3) = (3, -1). Getting (-1, -1) comes from shifting left instead of right, and (3, -7) comes from shifting down instead of up.

3. Question 3 · Inequalities

   What is the solution of the compound inequality -7 ≤ 3x - 1 < 11?
   • A.-8/3 ≤ x < 10/3
   • B.-2 ≤ x < 4
   • C.x ≤ -2 or x > 4
   • D.-2 < x ≤ 4
   Show answer and explanation

   Answer: B. -2 ≤ x < 4

   Add 1 to all three parts: -6 ≤ 3x < 12. Divide by 3: -2 ≤ x < 4. The answer -8/3 ≤ x < 10/3 comes from subtracting 1 instead of adding it, and an 'and' inequality gives values between the endpoints, not outside them.

4. Question 4 · Polynomials

   When (3x - 2)(x² + 4x - 5) is written in standard form, what is the coefficient of the x term?
   • A.23
   • B.-15
   • C.-23
   • D.-7
   Show answer and explanation

   Answer: C. -23

   The x terms come from 3x(-5) = -15x and -2(4x) = -8x, so the coefficient is -15 + (-8) = -23. The full product is 3x³ + 10x² - 23x + 10. Getting -7 comes from treating -2(4x) as +8x.

5. Question 5 · Quadratic Functions

   Which statement about the equation 4x² - 12x + 9 = 0 is true?
   • A.The discriminant is -288, so there are no real solutions.
   • B.The discriminant is 12, so there are two real solutions.
   • C.The discriminant is 288, so there are two real solutions.
   • D.The discriminant is 0, so there is exactly one real solution.
   Show answer and explanation

   Answer: D. The discriminant is 0, so there is exactly one real solution.

   b² - 4ac = (-12)² - 4(4)(9) = 144 - 144 = 0, so the equation has exactly one real solution (x = 3/2). The value 288 comes from adding 4ac instead of subtracting it.

6. Question 6 · Solving Equations

   Caleb's phone plan costs $18.00 per month plus $0.15 for each text message. His bill this month was $31.95. How many text messages did he send?
   • A.93
   • B.195
   • C.213
   • D.333
   Show answer and explanation

   Answer: A. 93

   Write 18 + 0.15t = 31.95. Subtract 18 to get 0.15t = 13.95, then divide: t = 93. Getting 213 comes from dividing the whole bill by 0.15, and 333 comes from adding $18.00 instead of subtracting it.

7. Question 7 · Functions and Sequences

   The value of a delivery van, in dollars, is modeled by V(t) = 18,000 - 1,500t, where t is the number of years since purchase and 0 ≤ t ≤ 8. What is the range of V for this domain?
   • A.0 ≤ V ≤ 18,000
   • B.0 ≤ V ≤ 8
   • C.1,500 ≤ V ≤ 18,000
   • D.6,000 ≤ V ≤ 18,000
   Show answer and explanation

   Answer: D. 6,000 ≤ V ≤ 18,000

   V decreases as t increases, so the largest value is V(0) = 18,000 and the smallest is V(8) = 18,000 - 12,000 = 6,000. The range is 6,000 ≤ V ≤ 18,000; 0 ≤ V ≤ 8 confuses the range with the domain, and 0 assumes the van loses all its value.

8. Question 8 · Inequalities

   What is the solution of 5 < 9 - 2x ≤ 17?
   • A.-13 ≤ x < -7
   • B.-4 ≤ x < 2
   • C.-4 < x ≤ 2
   • D.x < -4 or x ≥ 2
   Show answer and explanation

   Answer: B. -4 ≤ x < 2

   Subtract 9 from all parts: -4 < -2x ≤ 8. Divide by -2 and reverse both signs: 2 > x ≥ -4, which is -4 ≤ x < 2. Keeping the symbols in their original places gives -4 < x ≤ 2, and adding 9 instead of subtracting gives -13 ≤ x < -7.

9. Question 9 · Polynomials

   Which shows the product (x² - 4)(3x + 1) written in standard form?
   • A.3x³ + x² - 4
   • B.3x³ + x² - 12x - 4
   • C.3x³ - 11x² - 4
   • D.3x³ + x² - 12x + 4
   Show answer and explanation

   Answer: B. 3x³ + x² - 12x - 4

   Distribute: x²(3x + 1) = 3x³ + x² and -4(3x + 1) = -12x - 4. In descending order of exponents the product is 3x³ + x² - 12x - 4. The expression 3x³ + x² - 4 skips the product -4 · 3x.

10. Question 10 · Quadratic Functions

    Which equation has two distinct real solutions?
    • A.3x² + 2x - 1 = 0
    • B.2x² - 3x + 5 = 0
    • C.x² + 4x + 4 = 0
    • D.x² - 6x + 10 = 0
    Show answer and explanation

    Answer: A. 3x² + 2x - 1 = 0

    For 3x² + 2x - 1 = 0, the discriminant is 2² - 4(3)(-1) = 4 + 12 = 16 > 0, so there are two real solutions. The others have discriminants 16 - 16 = 0 (one solution), 9 - 40 = -31 and 36 - 40 = -4 (no real solutions).

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