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GA Milestones Algebra EOC study materials
GA Milestones Algebra EOC study resources

GA Milestones Algebra EOC practice questions

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Question 1Algebra

What is the slope of the line that passes through the points (2, 3) and (6, 11)?

  1. 8
  2. 1/2
  3. 2 Correct answer.
  4. 4

Explanation

Slope equals the change in y divided by the change in x, so (11 − 3) / (6 − 2) = 8 / 4 = 2. The line rises 2 units for every 1 unit it moves right.

Question 2Algebra

Which expression is equivalent to (3x^2y)(−2xy^3)?

  1. −6x^3y^4 Correct answer.
  2. −5x^3y^4
  3. −6x^2y^3
  4. 6x^3y^4

Explanation

Multiply the coefficients: 3 · (−2) = −6. Add the exponents of like bases: x^2 · x = x^3 and y · y^3 = y^4, giving −6x^3y^4.

Question 3Algebra

What is the solution set of the equation |2x − 5| = 9?

  1. {−2, 7} Correct answer.
  2. {2, 7}
  3. {7}
  4. {−7, 2}

Explanation

An absolute value equation splits into two cases: 2x − 5 = 9 gives x = 7, and 2x − 5 = −9 gives x = −2. Both values satisfy the original equation, so the solution set is {−2, 7}.

Question 4Algebra

The function f is defined by f(x) = 3x − 4. What is f(−2)?

  1. 10
  2. −10 Correct answer.
  3. −2
  4. 2

Explanation

Substitute −2 for x: 3(−2) − 4 = −6 − 4 = −10. Function notation means the input value replaces every x in the rule.

Question 5Algebra

Which equation represents a line parallel to y = (2/3)x + 5?

  1. y = −(2/3)x + 1
  2. y = −(3/2)x − 7
  3. y = (2/3)x − 7 Correct answer.
  4. y = (3/2)x + 5

Explanation

Parallel lines have equal slopes but different y-intercepts. The given line has slope 2/3, so y = (2/3)x − 7 is parallel to it because it shares that slope.

Question 6Algebra

Simplify (x^5)/(x^2) using the laws of exponents.

  1. x^7
  2. x^10
  3. x^3 Correct answer.
  4. x^2.5

Explanation

When dividing powers with the same base, subtract the exponents: x^5 / x^2 = x^(5−2) = x^3.

Question 7Algebra

What are the x-intercepts of the graph of y = x^2 − 5x + 6?

  1. x = 2 and x = 3 Correct answer.
  2. x = 1 and x = 6
  3. x = −1 and x = −6
  4. x = −2 and x = −3

Explanation

Factor the quadratic as (x − 2)(x − 3) = 0. Setting each factor equal to zero gives x = 2 and x = 3, the points where the graph crosses the x-axis.

Question 8Algebra

Which expression is the complete factorization of 4x^2 − 9?

  1. (4x + 3)(x − 3)
  2. (2x + 3)(2x − 3) Correct answer.
  3. (2x − 3)^2
  4. (2x − 9)(2x + 1)

Explanation

This is a difference of squares because 4x^2 = (2x)^2 and 9 = 3^2. The pattern a^2 − b^2 factors as (a + b)(a − b), giving (2x + 3)(2x − 3).

Question 9Algebra

Solve the system: 2x + y = 11 and x − y = 1. What is the value of x?

  1. 5
  2. 6
  3. 4 Correct answer.
  4. 3

Explanation

Adding the two equations eliminates y: 3x = 12, so x = 4. Substituting back gives y = 3, and both equations are satisfied.

Question 10Algebra

The quadratic function f(x) = x^2 − 6x + 5 has its vertex at which point?

  1. (−3, −4)
  2. (3, 4)
  3. (3, −4) Correct answer.
  4. (6, 5)

Explanation

The x-coordinate of the vertex is −b/(2a) = 6/2 = 3. Substituting x = 3 gives f(3) = 9 − 18 + 5 = −4, so the vertex is (3, −4).

Question 11Algebra

Which inequality represents all values of x for which 3x − 7 < 8?

  1. x > 1/3
  2. x < 5 Correct answer.
  3. x > 5
  4. x < 1/3

Explanation

Add 7 to both sides to get 3x < 15, then divide by the positive 3 to get x < 5. Dividing by a positive number does not flip the inequality sign.

Question 12Algebra

A line has a slope of −3 and passes through the point (2, 1). Which equation represents this line in point-slope form?

  1. y + 1 = −3(x + 2)
  2. y − 1 = 3(x − 2)
  3. y − 2 = −3(x − 1)
  4. y − 1 = −3(x − 2) Correct answer.

Explanation

Point-slope form is y − y1 = m(x − x1). Substituting m = −3, x1 = 2, and y1 = 1 gives y − 1 = −3(x − 2).

Question 13Algebra

If f(x) = x^2 + 1 and g(x) = 2x − 3, what is f(g(2))?

  1. 10
  2. 26
  3. 2 Correct answer.
  4. 5

Explanation

First evaluate the inner function: g(2) = 2(2) − 3 = 1. Then apply f to that result: f(1) = 1^2 + 1 = 2.

Question 14Algebra

Which expression is equivalent to (x + 4)(x − 4)?

  1. x^2 + 8x − 16
  2. x^2 − 8x + 16
  3. x^2 + 16
  4. x^2 − 16 Correct answer.

Explanation

This product is a difference of squares: (x + 4)(x − 4) = x^2 − 4^2 = x^2 − 16. The middle terms +4x and −4x cancel.

Question 15Algebra

The equation of a line is 6x + 2y = 10. What is the slope of this line?

  1. 3
  2. −3 Correct answer.
  3. 1/3
  4. −6

Explanation

Solve for y: 2y = −6x + 10, so y = −3x + 5. In slope-intercept form the coefficient of x is the slope, which is −3.

Question 16Algebra

A geometric sequence begins 3, 6, 12, 24, ... What is the 7th term of the sequence?

  1. 144
  2. 384
  3. 96
  4. 192 Correct answer.

Explanation

Each term is twice the previous term, so the common ratio is 2. Using a_n = a_1 · r^(n−1) gives 3 · 2^6 = 3 · 64 = 192.

Question 17Algebra

The expression (x^2 − 9)/(x + 3) is equivalent to which expression for x ≠ −3?

  1. x + 3
  2. x − 3 Correct answer.
  3. x^2 − 3
  4. x − 9

Explanation

Factor the numerator as (x + 3)(x − 3). The common factor (x + 3) cancels with the denominator, leaving x − 3 for x ≠ −3.

Question 18Algebra

What is the domain of the function f(x) = 1/(x − 4)?

  1. All real numbers except x = 4 Correct answer.
  2. All real numbers
  3. All real numbers greater than 4
  4. All real numbers except x = −4

Explanation

A rational function is undefined where its denominator equals zero. Setting x − 4 = 0 gives x = 4, so 4 must be excluded from the domain.

Question 19Algebra

A system of two linear equations has no solution. Which statement about the graphs of the equations must be true?

  1. The lines are the same line.
  2. The lines intersect at exactly one point.
  3. The lines are parallel and have the same slope but different y-intercepts. Correct answer.
  4. The lines are perpendicular to each other.

Explanation

No solution means the lines never intersect, which occurs only when they are parallel. Parallel lines share a slope but have different y-intercepts, so the equations are inconsistent.

Question 20Algebra

The graph of y = x^2 is shifted 3 units to the right and 2 units down. Which equation represents the transformed graph?

  1. y = (x + 3)^2 − 2
  2. y = (x − 3)^2 − 2 Correct answer.
  3. y = (x − 2)^2 − 3
  4. y = (x − 3)^2 + 2

Explanation

A horizontal shift right by h units replaces x with (x − h), giving (x − 3)^2. A vertical shift down by k units subtracts k, giving − 2.

Question 21Algebra

The function f(x) = 2x + 5 models the total cost, in dollars, of renting a machine for x hours. What does the slope of 2 represent in this context?

  1. The rental lasts 2 hours.
  2. The cost increases by $2 for each additional hour of rental. Correct answer.
  3. The cost increases by $5 for each additional hour.
  4. The total cost is $2 for any rental.

Explanation

In a linear model y = mx + b, the slope m gives the rate of change of y per unit of x. Here the slope 2 means cost rises $2 per additional hour.

Question 22Algebra

Solve x^2 − 4x − 12 = 0 by completing the square. Which equation results after completing the square?

  1. (x + 2)^2 = 16
  2. (x − 2)^2 = 16 Correct answer.
  3. (x − 4)^2 = 28
  4. (x − 2)^2 = 12

Explanation

Move the constant: x^2 − 4x = 12. Add (b/2)^2 = 4 to both sides to get x^2 − 4x + 4 = 16, which factors as (x − 2)^2 = 16.

Question 23Algebra

The quadratic function f(x) = 2x^2 − 8x + 1 has a minimum value. What is the minimum value of f?

  1. −3
  2. 1
  3. −1
  4. −7 Correct answer.

Explanation

The vertex x-coordinate is −b/(2a) = 8/4 = 2. Evaluating f(2) = 2(4) − 16 + 1 = −7 gives the minimum because a = 2 is positive, so the parabola opens upward.

Question 24Algebra

For what values of k does the equation x^2 + kx + 9 = 0 have exactly one real solution?

  1. k = 3 or k = −3
  2. k = 6 or k = −6 Correct answer.
  3. k = 9 or k = −9
  4. k = 0 only

Explanation

A quadratic has exactly one real solution when its discriminant b^2 − 4ac equals zero. Here k^2 − 4(1)(9) = 0, so k^2 = 36 and k = 6 or k = −6.

Question 25Algebra

The expression (3x^2 + 2x − 8) ÷ (x + 2) simplifies to which polynomial?

  1. 3x + 4
  2. 3x − 4 Correct answer.
  3. 3x − 8
  4. 3x^2 − 4

Explanation

Polynomial division gives 3x − 4, and multiplying back checks: (x + 2)(3x − 4) = 3x^2 + 2x − 8. Alternatively, the numerator factors as (x + 2)(3x − 4).

Question 26Algebra

A function is defined as f(x) = 3x − 7. What is f(4)?

  1. 19
  2. 5 Correct answer.
  3. −5
  4. 12

Explanation

Substituting x = 4 gives f(4) = 3(4) − 7 = 12 − 7 = 5. The order of operations requires multiplying before subtracting.

Question 27Algebra

Which expression represents the slope of the line passing through the points (1, 5) and (4, 11)?

  1. (4 − 1)/(11 − 5)
  2. (11 − 5)/(4 − 1) Correct answer.
  3. (11 − 4)/(5 − 1)
  4. (11 + 5)/(4 + 1)

Explanation

Slope is the change in y divided by the change in x, so m = (y₂ − y₁)/(x₂ − x₁) = (11 − 5)/(4 − 1). This equals 6/3 = 2.

Question 28Algebra

What is the value of |−8| + |3|?

  1. 24
  2. 11 Correct answer.
  3. −5
  4. 5

Explanation

Absolute value gives the distance from zero, so |−8| = 8 and |3| = 3. Their sum is 8 + 3 = 11.

Question 29Algebra

Which expression is equivalent to 2(3x − 4) + 5x?

  1. 11x − 4
  2. 16x
  3. 11x − 8 Correct answer.
  4. 6x − 8

Explanation

Distributing gives 6x − 8, and combining like terms with 5x gives 11x − 8. Only the x-terms combine; the constant stays separate.

Question 30Algebra

The graph of a linear function passes through (0, −2) and has a slope of 4. Which equation represents the function?

  1. y = 4x + 2
  2. y = −2x + 4
  3. y = 2x − 4
  4. y = 4x − 2 Correct answer.

Explanation

The point (0, −2) is the y-intercept, so b = −2, and the slope is m = 4. Slope-intercept form y = mx + b gives y = 4x − 2.

Question 31Algebra

Which value of x satisfies the equation 5x − 3 = 2x + 12?

  1. 9
  2. 5 Correct answer.
  3. 3
  4. 15

Explanation

Subtracting 2x from both sides gives 3x − 3 = 12, and adding 3 gives 3x = 15, so x = 5. Checking: 5(5) − 3 = 22 and 2(5) + 12 = 22.

Question 32Algebra

The sequence 7, 11, 15, 19, ... is arithmetic. Which expression gives the nth term of the sequence?

  1. 7n
  2. n + 4
  3. 4n + 3 Correct answer.
  4. 4n + 7

Explanation

The common difference is 4 and the first term is 7, so aₙ = 7 + (n − 1)4 = 4n + 3. Checking n = 1 gives 7 and n = 2 gives 11.

Question 33Algebra

What is the solution set of the inequality −2x ≥ 10?

  1. x ≥ −5
  2. x ≤ −5 Correct answer.
  3. x ≥ 5
  4. x ≤ 5

Explanation

Dividing both sides by −2 requires reversing the inequality sign, giving x ≤ −5. Failing to reverse the sign is the most common error.

Question 34Algebra

Which expression is equivalent to (2x^3 y^2)(3x^2 y^4)?

  1. 6x^5 y^6 Correct answer.
  2. 6x^6 y^8
  3. 5x^5 y^6
  4. 6x^5 y^8

Explanation

Multiply the coefficients 2 · 3 = 6 and add exponents of like bases: x^(3+2) = x^5 and y^(2+4) = y^6. The product is 6x^5 y^6.

Question 35Algebra

Which expression is the factored form of x^2 + 7x + 12?

  1. (x + 1)(x + 12)
  2. (x + 2)(x + 6)
  3. (x − 3)(x − 4)
  4. (x + 3)(x + 4) Correct answer.

Explanation

We need two numbers whose product is 12 and whose sum is 7, namely 3 and 4. So x^2 + 7x + 12 = (x + 3)(x + 4).

Question 36Algebra

The function f(x) = x^2 + 4x − 5 has a graph with x-intercepts at which pair of values?

  1. 5 and 5
  2. 2 and −2
  3. 1 and −5 Correct answer.
  4. −1 and 5

Explanation

Setting x^2 + 4x − 5 = 0 and factoring gives (x + 5)(x − 1) = 0, so x = −5 or x = 1. These are the values where the graph crosses the x-axis.

Question 37Algebra

Which expression is equivalent to (x^2 − 4x − 21)/(x − 7) for x ≠ 7?

  1. x^2 − 3
  2. x + 3 Correct answer.
  3. x − 3
  4. x + 7

Explanation

The numerator factors as (x − 7)(x + 3), so the factor (x − 7) cancels with the denominator. The simplified quotient is x + 3.

Question 38Algebra

The system 3x + 2y = 12 and 6x + 4y = 18 is graphed. Which statement describes the solution set?

  1. There are exactly two solutions because the lines intersect twice.
  2. There is no solution because the lines are parallel. Correct answer.
  3. There are infinitely many solutions because the equations are equivalent.
  4. There is exactly one solution because the slopes differ.

Explanation

The second equation is 2 times the left side of the first but not 2 times the constant, so the lines have the same slope and different intercepts. Parallel lines never intersect, so there is no solution.

Question 39Algebra

The function f(x) = −(x − 2)^2 + 9 models the height of a ball. What is the maximum value of f?

  1. 11
  2. −2
  3. 9 Correct answer.
  4. 2

Explanation

The function is in vertex form a(x − h)^2 + k with a = −1, h = 2, and k = 9. Because a is negative, the vertex (2, 9) is a maximum, so the maximum value is 9.

Question 40Algebra

If f(x) = x + 3 and g(x) = x^2, which expression represents f(g(x))?

  1. x^2 + 9
  2. x^2 + 3x
  3. (x + 3)^2
  4. x^2 + 3 Correct answer.

Explanation

In f(g(x)), the output of g becomes the input of f. Substituting g(x) = x^2 into f gives f(x^2) = x^2 + 3.

Question 41Algebra

The expression √50 + √18 simplifies to which value?

  1. 2√17
  2. 34
  3. √68
  4. 8√2 Correct answer.

Explanation

Rewrite each radical using perfect-square factors: √50 = 5√2 and √18 = 3√2. Adding like radicals gives 5√2 + 3√2 = 8√2.

Question 42Algebra

A line passes through (3, −1) and is perpendicular to the line y = (2/3)x + 5. Which equation represents this line?

  1. y = (−2/3)x + 1
  2. y = (3/2)x − 11/2
  3. y = (−3/2)x + 7/2 Correct answer.
  4. y = (2/3)x − 3

Explanation

Perpendicular slopes are negative reciprocals, so the new slope is −3/2. Using point-slope form with (3, −1): y + 1 = (−3/2)(x − 3), which simplifies to y = (−3/2)x + 7/2.

Question 43Algebra

The quadratic equation 2x^2 − 5x + 1 = 0 has roots that can be found using the quadratic formula. Which expression gives the discriminant?

  1. 25
  2. −17
  3. 17 Correct answer.
  4. 33

Explanation

The discriminant is b^2 − 4ac. Here a = 2, b = −5, and c = 1, so (−5)^2 − 4(2)(1) = 25 − 8 = 17. A positive discriminant means two distinct real roots.

Question 44Algebra

A quantity grows according to the model P(t) = 500(1.06)^t, where t is in years. Which statement best describes this model?

  1. The quantity increases by 6% each year. Correct answer.
  2. The quantity increases by 0.06 units each year.
  3. The quantity increases by 6 units each year.
  4. The quantity decreases by 6% each year.

Explanation

In an exponential model P(t) = a(1 + r)^t, the base 1.06 means the growth rate r is 0.06, or 6% per year. The initial value is 500.

Question 45Algebra

The function f(x) = x^2 − 8x + 12 is rewritten in vertex form. Which expression is equivalent?

  1. (x − 4)^2 + 12
  2. (x − 4)^2 − 4 Correct answer.
  3. (x + 4)^2 − 4
  4. (x − 8)^2 − 52

Explanation

Complete the square: x^2 − 8x + 12 = (x^2 − 8x + 16) − 16 + 12 = (x − 4)^2 − 4. The vertex is (4, −4).

Question 46Algebra

The polynomial p(x) = x^3 − 2x^2 − 5x + 6 has x = 1 as a zero. Which expression represents p(x) completely factored?

  1. (x − 1)(x − 2)(x + 3)
  2. (x − 1)(x^2 − 5x + 6)
  3. (x − 1)(x + 2)(x − 3) Correct answer.
  4. (x + 1)(x − 2)(x − 3)

Explanation

Dividing by (x − 1) gives x^2 − x − 6, which factors as (x − 3)(x + 2). Thus p(x) = (x − 1)(x + 2)(x − 3).

Question 47Algebra

The rational function f(x) = (x^2 − 1)/(x^2 − 4) is graphed. Which statement is true about its asymptotes?

  1. It has vertical asymptotes at x = 2 and x = −2 and a horizontal asymptote at y = 1. Correct answer.
  2. It has vertical asymptotes at x = 1 and x = −1 and a horizontal asymptote at y = 1.
  3. It has vertical asymptotes at x = 2 and x = −2 and no horizontal asymptote.
  4. It has a horizontal asymptote at y = 0 and a hole at x = 2.

Explanation

The denominator factors as (x − 2)(x + 2), so vertical asymptotes occur at x = ±2. The numerator and denominator have equal degree, so the horizontal asymptote is the ratio of leading coefficients, y = 1/1 = 1.

Question 48Algebra

A system consists of y = x^2 − 4x + 5 and y = 2x − 4. What is the number of real solutions to this system?

  1. 2
  2. Infinitely many
  3. 1 Correct answer.
  4. 0

Explanation

Setting the expressions equal gives x^2 − 4x + 5 = 2x − 4, so x^2 − 6x + 9 = 0, which is (x − 3)^2 = 0. There is one repeated root, so the line is tangent to the parabola and the system has exactly one real solution.

Question 49Algebra

The function f(x) = 3x + 2 is transformed to g(x) = f(x − 4) − 1. How does the graph of g compare with the graph of f?

  1. It shifts 4 units right and 1 unit down. Correct answer.
  2. It shifts 4 units right and 1 unit up.
  3. It shifts 1 unit right and 4 units down.
  4. It shifts 4 units left and 1 unit down.

Explanation

Replacing x with x − 4 shifts the graph 4 units to the right, and subtracting 1 outside the function shifts it 1 unit down. So g(x) = 3(x − 4) + 2 − 1 = 3x − 11.

Question 50Algebra

The equation x^2 + y^2 − 6x + 10y + 18 = 0 represents a circle. What are the center and radius of this circle?

  1. Center (3, −5), radius 16
  2. Center (3, −5), radius 4 Correct answer.
  3. Center (−3, 5), radius 4
  4. Center (6, −10), radius 18

Explanation

Complete the square for each variable: (x^2 − 6x + 9) + (y^2 + 10y + 25) = −18 + 9 + 25 = 16. This gives (x − 3)^2 + (y + 5)^2 = 16, so the center is (3, −5) and the radius is √16 = 4.