Round 1 Review Sheet Round 1 Review Sheet
Regents Power Prep: Equations, Expressions, & Functions
Student Name:
Date:
Part I: Multiple Choice
(2 credits each)
1.The volume of a cone is given by the formula \(V = \frac{1}{3}Bh\). Which equation is correctly solved for \(B\)?
(1) \(B = \frac{V}{3h}\)
(2) \(B = \frac{3V}{h}\)
(3) \(B = 3Vh\)
(4) \(B = \frac{h}{3V}\)
2.The expression \(x^2 - 100\) is equivalent to which of the following?
(1) \((x - 10)(x - 10)\)
(2) \((x + 10)(x + 10)\)
(3) \((x - 50)(x + 50)\)
(4) \((x - 10)(x + 10)\)
3.Which graph represents a function with a range of \(y \geq -4\)?
(1)
(2)
(3)
(4)
4.What is the solution to the inequality \(3(x - 4) > 5x + 2\)?
(1) \(x > 7\)
(2) \(x < 7\)
(3) \(x > -7\)
(4) \(x < -7\)
5.If \(f(x) = 2x^2 - 1\), what is the value of \(f(-3)\)?
(1) \(-19\)
(2) \(11\)
(3) \(17\)
(4) \(-13\)
6.A movie theater charges $12.50 per adult ticket (\(a\)) and $8.25 per child ticket (\(c\)). A family spent $74.00 on tickets for 7 people. Which system of equations represents this situation?
(1) \(a + c = 74\) and \(12.50a + 8.25c = 7\)
(2) \(a + c = 7\) and \(12.50a + 8.25c = 74\)
(3) \(a + c = 7\) and \(8.25a + 12.50c = 74\)
(4) \(a - c = 7\) and \(12.50a + 8.25c = 74\)
Part II: Constructed Response
(Points vary)
7.Express the sum of \((3x^2 - 4x + 7)\) and \((2x^2 + 5x - 8)\) as a trinomial in standard form.
8.Solve the following equation algebraically for \(x\):
\(4(2x - 1) = 6x + 8\)
9.The table below shows the cost of renting a bicycle for various hours. State the average rate of change, in dollars per hour, for the interval between 2 and 5 hours.
Hours (\(x\)) Cost (\(y\)) 1 $15 2 $25 3 $35 5 $55
10. Case Study: Error Analysis
A student was asked to solve the equation: \(2x + 10 = 4x - 6\). Their work is shown below.
Step 1: \(2x + 10 = 4x - 6\)
Step 2: \(10 = 2x - 6\)
Step 3: \(4 = 2x\)
Step 4: \(x = 2\)
Identify the step where the student made an error and explain what they did wrong. Then, find the correct solution.
Blueprint Review Series • Algebra I NYS Regents Prep • Round 1 of 5
Round 1 Answer Key Teacher Resource
Round 1 Answer Key
Equation Essentials Review • Scoring Guide
Solution Set
Part I: Multiple Choice Solutions
1
Answer: (2) \(B = \frac{3V}{h}\)
Logic: Multiply both sides by 3 to get \(3V = Bh\), then divide by \(h\).
2
Answer: (4) \((x - 10)(x + 10)\)
Logic: Difference of two squares pattern: \(a^2 - b^2 = (a-b)(a+b)\).
3
Answer: (2)
Logic: Option (2) shows a parabola with a vertex at \((0, -4)\) opening upward, meaning all \(y\)-values are \(-4\) or higher.
4
Answer: (4) \(x < -7\)
Logic: \(3x - 12 > 5x + 2 \rightarrow -14 > 2x \rightarrow -7 > x\). Remember to flip if dividing by negative, but here we divided by positive 2. \(x < -7\).
5
Answer: (3) \(17\)
Logic: \(2(-3)^2 - 1 = 2(9) - 1 = 18 - 1 = 17\). Distractor (1) comes from squaring incorrectly as \(-9\).
6
Answer: (2)
Logic: \(a + c = 7\) (quantity) and \(12.50a + 8.25c = 74\) (value/cost).
Part II: Constructed Response Solutions
7. Sum as a Trinomial
\(5x^2 + x - 1\)
Combine like terms: \((3x^2 + 2x^2) + (-4x + 5x) + (7 - 8)\)
8. Algebraic Solution
\(8x - 4 = 6x + 8\)
\(2x - 4 = 8\)
\(2x = 12\)
\(x = 6\)
9. Average Rate of Change
\( \frac{55 - 25}{5 - 2} = \frac{30}{3} = 10 \text{ dollars per hour}\)
10. Error Analysis Key
Identification:
The student made an error in Step 3.
Explanation:
In Step 2, the equation was \(10 = 2x - 6\). To isolate \(2x\), the student should have added 6 to both sides, resulting in \(16 = 2x\). Instead, it appears they subtracted 6 from 10, incorrectly getting 4.
Correct Solution:
\(16 = 2x \rightarrow \mathbf{x = 8}\)
Teacher Answer Key • Round 1: Equations • Blueprint Series
Round 2 Review Sheet Round 2 Review Sheet
Regents Power Prep: Expression Experts (Polynomials & Factoring)
Student Name:
Date:
Part I: Multiple Choice
(2 credits each)
1.The expression \(2x(x - 4) - 3(x + 5)\) is equivalent to
(1) \(2x^2 - 11x - 15\)
(2) \(2x^2 - 11x + 15\)
(3) \(2x^2 - 5x - 15\)
(4) \(2x^2 - 11x - 5\)
2.Which expression represents \(x^2 - 7x - 18\) factored completely?
(1) \((x - 9)(x + 2)\)
(2) \((x + 9)(x - 2)\)
(3) \((x - 6)(x - 3)\)
(4) \((x - 9)(x - 2)\)
3.The expression \(36x^2 - 49\) is equivalent to
(1) \((6x - 7)(6x - 7)\)
(2) \((6x + 7)(6x + 7)\)
(3) \((6x - 7)(6x + 7)\)
(4) \((18x - 7)(18x + 7)\)
4.What is the greatest common factor (GCF) of the expression \(12x^4y^2 - 18x^2y^3\)?
(1) \(6x^2y^2\)
(2) \(6x^4y^3\)
(3) \(3x^2y^2\)
(4) \(2x^2y^2\)
5.Which expression is not equivalent to \((x + 3)^2\)?
(1) \(x^2 + 9\)
(2) \(x^2 + 6x + 9\)
(3) \((x + 3)(x + 3)\)
(4) \(x^2 + 3x + 3x + 9\)
6.The length of a rectangle is represented by \(2x + 5\) and the width is represented by \(x - 3\). Which expression represents the area of the rectangle?
(1) \(2x^2 - 15\)
(2) \(2x^2 - x - 15\)
(3) \(2x^2 + x - 15\)
(4) \(3x + 2\)
Part II: Constructed Response
(Points vary)
7.Multiply \((x - 4)\) and \((3x^2 - 2x + 5)\). Express your answer as a polynomial in standard form.
8.Factor the following trinomial completely:
\(3x^2 + 13x - 10\)
9.Factor the expression \(4x^2 - 100\) completely.
10. Case Study: Factoring Error
A student was asked to factor the expression \(x^2 - 10x + 16\). Their work is shown below.
Expression: \(x^2 - 10x + 16\)
Factors of 16: 1, 16 | 2, 8 | 4, 4
Student's Result: \((x - 8)(x + 2)\)
Explain why the student's result is incorrect. Then, provide the correct factored form of the expression.
Blueprint Review Series • Algebra I NYS Regents Prep • Round 2 of 5
Round 2 Answer Key Teacher Resource
Round 2 Answer Key
Expression Experts Review • Scoring Guide
Solution Set
Part I: Multiple Choice Solutions
1
Answer: (1) \(2x^2 - 11x - 15\)
Logic: \(2x^2 - 8x - 3x - 15 = 2x^2 - 11x - 15\). Watch the negative distribution on the second term.
2
Answer: (1) \((x - 9)(x + 2)\)
Logic: Find two numbers that multiply to \(-18\) and add to \(-7\). \(-9\) and \(2\) work.
3
Answer: (3) \((6x - 7)(6x + 7)\)
Logic: Difference of squares: \((6x)^2 - (7)^2\).
4
Answer: (1) \(6x^2y^2\)
Logic: Numerical GCF is 6. Variable GCF uses lowest powers: \(x^2\) and \(y^2\).
5
Answer: (1) \(x^2 + 9\)
Logic: \((x+3)^2\) results in a middle term \(6x\). Option (1) is a common "freshman's dream" error.
6
Answer: (2) \(2x^2 - x - 15\)
Logic: \((2x+5)(x-3) = 2x^2 - 6x + 5x - 15\). Simplify to get \(-x\) in middle.
Part II: Constructed Response Solutions
7. Multiply Binomial by Trinomial
\(3x^3 - 14x^2 + 13x - 20\)
Process: \(x(3x^2 - 2x + 5) - 4(3x^2 - 2x + 5) = (3x^3 - 2x^2 + 5x) + (-12x^2 + 8x - 20)\)
8. Factor Completely (\(a > 1\))
\((3x - 2)(x + 5)\)
Split middle: \(3x^2 + 15x - 2x - 10 \rightarrow 3x(x+5) - 2(x+5)\)
9. Factor Completely (Multi-Step)
\(4(x - 5)(x + 5)\)
Factor GCF first: \(4(x^2 - 25)\), then difference of squares.
10. Factoring Error Key
Explanation:
The student correctly identified that \(8\) and \(2\) are factors of \(16\), but their signs are incorrect. A product of \(+16\) requires either two positives or two negatives. Since the sum must be \(-10\), both factors must be negative (\(-8\) and \(-2\)). The student's answer \((x-8)(x+2)\) would multiply to \(-16\).
Correct Factored Form:
\((x - 8)(x - 2)\)
Teacher Answer Key • Round 2: Expressions • Blueprint Series
Round 3 Review Sheet Round 3 Review Sheet
Regents Power Prep: Inequality Icons (Inequalities & Systems)
Student Name:
Date:
Part I: Multiple Choice
(2 credits each)
1.What is the solution to the inequality \(-2(x - 5) \leq 12\)?
(1) \(x \leq -1\)
(2) \(x \geq -1\)
(3) \(x \leq 11\)
(4) \(x \geq -11\)
2.Which point is in the solution set of the system of inequalities graphed below?
(1) \((0, 0)\)
(2) \((5, 5)\)
(3) \((-5, 0)\)
(4) \((0, 5)\)
3.An online store sells t-shirts for $15 each and hats for $10 each. A customer wants to spend at most $100. Which inequality represents this situation, where \(t\) is t-shirts and \(h\) is hats?
(1) \(15t + 10h \geq 100\)
(2) \(15t + 10h \leq 100\)
(3) \(10t + 15h \leq 100\)
(4) \(15t + 10h < 100\)
4.Which system of equations has the same solution as the system below?
\(x + y = 10\)
(1) \(3x = 18\)
(2) \(x = 2\)
(3) \(3x = 2\)
(4) \(3x = 10\)
5.The solution to a system of linear equations is \((2, -3)\). Which of the following equations could be part of that system?
(1) \(x + y = 5\)
(2) \(2x - y = 7\)
(3) \(y = 3x - 9\)
(4) \(y = -2x + 1\)
6.Which inequality is shown by the graph where the boundary line is \(y = 2x - 4\), the line is solid, and the shading is above the line?
(1) \(y \geq 2x - 4\)
(2) \(y > 2x - 4\)
(3) \(y \leq 2x - 4\)
(4) \(y < 2x - 4\)
Part II: Constructed Response
(Points vary)
7.Solve the following system of equations algebraically:
\(y = 3x - 5\)
8.Graph the following system of inequalities on the grid below. Label the solution set with an S .
\(y < -2x + 4\)
9.Marcus wants to buy a game console that costs $350. He has $120 saved and plans to save $25 per week. Write and solve an inequality to find the minimum number of full weeks Marcus needs to save.
10. Case Study: Inequality Signs
A student was asked to solve the inequality \(-4x + 8 > 24\). Their work is shown below.
Step 1: \(-4x + 8 > 24\)
Step 2: \(-4x > 16\)
Step 3: \(x > -4\)
Identify the error in the student's work and explain what they should have done differently. State the correct solution.
Round 3 Answer Key Teacher Resource
Round 3 Answer Key
Inequality Icons Review • Scoring Guide
Solution Set
Part I: Multiple Choice Solutions
1
Answer: (2) \(x \geq -1\)
Logic: \(-2x + 10 \leq 12 \rightarrow -2x \leq 2\). When dividing by \(-2\), flip the sign: \(x \geq -1\).
2
Answer: (3) \((-5, 0)\)
Logic: Point \((-5, 0)\) lies within the overlapping shaded region of the two lines.
3
Answer: (2) \(15t + 10h \leq 100\)
Logic: "At most" indicates \(\leq\). T-shirts are $15, hats are $10.
4
Answer: (1) \(3x = 18\)
Logic: Using the elimination method, add the two equations: \((x + 2x) + (y - y) = 10 + 8 \rightarrow 3x = 18\).
5
Answer: (2) \(2x - y = 7\)
Logic: Substitute \((2, -3)\): \(2(2) - (-3) = 4 + 3 = 7\). This makes the equation true.
6
Answer: (1) \(y \geq 2x - 4\)
Logic: Solid line means \(\geq\) or \(\leq\). Shading above means \(\geq\).
Part II: Constructed Response Solutions
7. Solve System Algebraically
\(x = 4, y = 7\) or \((4, 7)\)
Substitution: \(2x + 3(3x - 5) = 29 \rightarrow 2x + 9x - 15 = 29 \rightarrow 11x = 44 \rightarrow x = 4\). Plug in to find \(y\).
8. Graph System
Line 1: \(y\)-int of 4, slope -2, dashed line, shade below.S is the intersection region.
9. Word Problem Modeling
\(120 + 25w \geq 350\)
\(25w \geq 230 \rightarrow w \geq 9.2\). Marcus needs 10 full weeks.
10. Inequality Sign Error Key
Explanation:
The student correctly subtracted 8 from both sides in Step 2, but in Step 3 they failed to flip the inequality sign when dividing by a negative number (\(-4\)). Anytime an inequality is multiplied or divided by a negative, the sign must change direction.
Correct Solution:
\(x < -4\)
Teacher Answer Key • Round 3: Inequalities • Blueprint Series
Round 4 Review Sheet Round 4 Review Sheet
Regents Power Prep: Function Figures (Notation & Interpretation)
Student Name:
Date:
Part I: Multiple Choice
(2 credits each)
1.If \(f(n) = (n - 1)^2 + 3n\), what is the value of \(f(-2)\)?
(1) \(3\)
(2) \(15\)
(3) \(-1\)
(4) \(5\)
2.An online store sells songs for $0.99 each. Which domain would be most appropriate for a function that calculates the total cost to download \(x\) songs?
(1) rational numbers
(2) integers
(3) whole numbers
(4) real numbers
3.The range of the function \(f(x) = x^2 + 2x - 8\) is all real numbers
(1) less than or equal to -9
(2) greater than or equal to -9
(3) less than or equal to -1
(4) greater than or equal to -1
4.Which representation does not describe a function?
(1)
\(\{(1,2), (2,3), (3,4)\}\)
(2)
A, B
1, 2
(3)
(4)
5.The value of the \(y\)-intercept of the function \(g(x) = 2x - 5\) is
(1) \(2\)
(2) \(-5\)
(3) \(2.5\)
(4) \(0\)
6.Which function has the greatest average rate of change over the interval \(0 \leq x \leq 2\)?
(1) \(f(x) = 3x + 1\)
(2) \(g(x) = x^2 + 1\)
(3) \(h(x)\) is shown by the table: \((0,1), (1,5), (2,10)\)
(4) \(k(x)\) is shown by the points: \((0,4), (1,5), (2,6)\)
Part II: Constructed Response
(Points vary)
7.A function is graphed on the set of axes below. State the domain and range of this function.
Domain:
Range:
8.State the zeros of the function \(f(x) = x^2 - 5x - 24\). Show all algebraic work.
9.The table below shows the temperature of a liquid over time. State the average rate of change, in degrees per minute, for the interval \(10 \leq t \leq 25\).
Round 4 Answer Key Teacher Resource
Round 4 Answer Key
Function Figures Review • Scoring Guide
Solution Set
Part I: Multiple Choice Solutions
1
Answer: (1) \(3\)
Logic: \(f(-2) = (-2 - 1)^2 + 3(-2) = (-3)^2 - 6 = 9 - 6 = 3\).
2
Answer: (3) whole numbers
Logic: You can only download non-negative, integer amounts of songs (0, 1, 2...).
3
Answer: (2) greater than or equal to -9
Logic: Vertex at \(x = -1\). \(f(-1) = (-1)^2 + 2(-1) - 8 = 1 - 2 - 8 = -9\). Parabola opens up.
4
Answer: (1)
Logic: In table (1), the input \(x=1\) is mapped to two different outputs (\(5\) and \(15\)).
5
Answer: (2) \(-5\)
Logic: Set \(x = 0\). \(g(0) = 2(0) - 5 = -5\). This is the constant term.
6
Answer: (3) \(h(x)\)
Logic: Average rates: (1) 3, (2) 2, (3) \((10-1)/(2-0) = 4.5\), (4) 1. \(h(x)\) is the steepest.
Part II: Constructed Response Solutions
7. Domain and Range
Domain: \(-5 < x \leq 5\) or \((-5, 5]\)
Range: \(0 \leq y \leq 4\) or \([0, 4]\)
8. Zeros of a Function
\(x = 8, x = -3\)
Factor: \((x - 8)(x + 3) = 0\). Solve each factor for zero.
9. Rate of Change
\( \frac{50 - 80}{25 - 10} = \frac{-30}{15} = -2 \text{ degrees per min}\)
10. Notation Evaluation Key
Explanation:
The student made an order of operations error in Step 2. When evaluating \(-x^2\), you must square the value first and then apply the negative sign. \((-3)^2\) is \(9\), so \(-x^2\) becomes \(-(9)\), which is \(-9\). The student likely treated it as \((-3)^2\) without the leading negative or incorrectly distributed it.
Correct Solution:
\(f(-3) = -(-3)^2 + 4 = -9 + 4 = \mathbf{-5}\)
Teacher Answer Key • Round 4: Functions • Blueprint Series
Round 5 Review Sheet Round 5: Ultimate Challenge
Regents Power Prep: Mixed Cumulative Mastery
Student Name:
Date:
Part I: Multiple Choice
(2 credits each)
1.When the equation \(\frac{x-1}{2} - \frac{a}{4} = \frac{3a}{4}\) is solved for \(x\) in terms of \(a\), the solution is
(1) \(2a + 1\)
(2) \(a + 1\)
(3) \(4a + 1\)
(4) \(\frac{3a}{2} + 1\)
2.Subtract \(3x^2 - 4x + 7\) from \(5x^2 + 2x - 3\). The result is
(1) \(2x^2 + 6x - 10\)
(2) \(2x^2 - 2x + 4\)
(3) \(8x^2 - 2x + 4\)
(4) \(2x^2 + 6x + 4\)
3.The expression \(2x^2 - 10x - 48\) factored completely is
(1) \(2(x - 8)(x + 3)\)
(2) \(2(x + 8)(x - 3)\)
(3) \((2x - 16)(x + 3)\)
(4) \(2(x^2 - 5x - 24)\)
4.A system of inequalities is graphed below. Which point is not in the solution set?
(1) \((4, 2)\)
(2) \((5, -1)\)
(3) \((0, 5)\)
(4) \((2, 3)\)
5.The cost of a pizza is $12 plus $1.50 per topping (\(t\)). Which function represents the total cost, \(C(t)\)?
(1) \(C(t) = 12t + 1.50\)
(2) \(C(t) = 1.50t + 12\)
(3) \(C(t) = 13.50t\)
(4) \(C(t) = 1.50(t + 12)\)
6.Function \(A\) is defined by \(y = 4x + 2\). Function \(B\) passes through \((0, 5)\) and \((2, 9)\). Which statement is true?
(1) Function A has a greater slope.
(2) Function B has a greater slope.
(3) Both functions have the same slope.
(4) Both functions have the same y-intercept.
Part II: Constructed Response
(Points vary)
7.Solve the equation algebraically for \(x\):
\(3(x - 2) + 4 = 5x - 12\)
8.For a class picnic, two teachers purchased drinks. One teacher bought 18 juice boxes and 32 bottles of water for $19.92. The other bought 14 juice boxes and 26 bottles of water for $15.76. Write and solve a system of equations to find the cost of each juice box and each bottle of water.
9.Factor \(x^4 - 81\) completely.
10. Final Case: Combined Error
A student was asked to simplify: \(2(3x + 5) - (4x - 2)\). Their work is shown below.
Step 1: \(6x + 10 - 4x - 2\)
Step 2: \(2x + 8\)
Identify the error in Step 1. Then, provide the correct simplified expression.
Blueprint Review Series • Algebra I NYS Regents Prep • Round 5 of 5
Round 5 Answer Key Teacher Resource
Round 5 Answer Key
Ultimate Challenge Review • Scoring Guide
Solution Set
Part I: Multiple Choice Solutions
1
Answer: (1) \(2a + 1\)
Logic: Clear fractions by multiplying by 4: \(2(x-1) - a = 3a \rightarrow 2x - 2 = 4a \rightarrow 2x = 4a + 2 \rightarrow x = 2a + 1\).
2
Answer: (1) \(2x^2 + 6x - 10\)
Logic: Distribute negative to terms being subtracted: \(5x^2 - 3x^2 + 2x - (-4x) - 3 - 7\).
3
Answer: (1) \(2(x - 8)(x + 3)\)
Logic: GCF of 2 first: \(2(x^2 - 5x - 24)\). Factors of \(-24\) that add to \(-5\) are \(-8\) and \(3\).
4
Answer: (3) \((0, 5)\)
Logic: The point \((0, 5)\) lies exactly on the dashed boundary line \(y = -x + 5\). Since the inequality is strictly greater than (\(>\)), points on the line are not included.
5
Answer: (2) \(C(t) = 1.50t + 12\)
Logic: \(1.50\) is the rate (per topping), and \(12\) is the fixed start value.
6
Answer: (1) Function A has a greater slope.
Logic: Slope A = 4. Slope B = \((9-5)/(2-0) = 2\). \(4 > 2\).
Part II: Constructed Response Solutions
7. Solve Multi-Step Equation
\(x = 5\)
\(3x - 6 + 4 = 5x - 12 \rightarrow 3x - 2 = 5x - 12 \rightarrow 10 = 2x\)
8. Systems Word Problem
\(18j + 32w = 19.92\)
Cost of Juice Box (\(j\)): $0.68
9. Factor Completely
\((x - 3)(x + 3)(x^2 + 9)\)
First: \((x^2 - 9)(x^2 + 9)\). Then factor the difference of squares further.
10. Combined Error Analysis Key
Identification:
The error is in the distribution of the negative sign in Step 1.
Explanation:
When distributing the negative sign across the second binomial \((4x - 2)\), the student failed to apply the negative to the \(-2\). Subtracting a negative is the same as adding, so it should have resulted in \(+2\), not \(-2\).
Correct Simplified Expression:
\(6x + 10 - 4x + 2 = \mathbf{2x + 12}\)
Teacher Answer Key • Round 5: Mastery • Blueprint Series
