Week 1 Power Sprint Worksheet Regents Power Sprint
Week 1: Numbers & Expressions
Name:
Date:
Section 1: The Basics (2 Credits Each)
Question 1 NYS June '25
The expression \(\sqrt{50}\) is equivalent to
1 \(5\)
2 \(\sqrt{20}\)
3 \(5\sqrt{2}\)
4 \(10\sqrt{2}\)
Question 2 (Q5) NYS June '25
Which polynomial has a degree of 3 and a leading coefficient of 2?
1 \(2x^2 + 3x + 1\)
2 \(6x^3 + 3x^2 - 2x\)
3 \(3x^2 + 2x + 2\)
4 \(2x^3 + x^2 + 4x\)
Question 3 (Q6) NYS June '25
The expression \((-3x^2 + 9) - (7x^2 - 5x + 4)\) is equivalent to
1 \(-10x^2 + 5x + 5\)
2 \(-10x^2 + 5x + 13\)
3 \(-10x^2 - 5x + 5\)
4 \(-10x^2 - 5x + 13\)
Section 2: Number Sense & Factoring
Question 4 (Q14)
The sum of \(\sqrt{3}\) and \(\sqrt{5}\) is
1
rational, since the sum can be expressed as an integer
2
rational, since the sum can be expressed as a nonterminating decimal
3
irrational, since the sum can be expressed as a terminating decimal
4
irrational, since the sum cannot be expressed as a terminating or repeating decimal
Question 5 (Q16)
Show Your Work Below
The sum of \(2\sqrt{27}\) and \(4\sqrt{12}\) is
1 \(14\sqrt{3}\)
2 \(34\sqrt{3}\)
3 \(6\sqrt{39}\)
4 \(8\sqrt{39}\)
Section 3: Free Response Performance
Question 6 (Q27) 2 CREDITS
Express \((5x - 3)(-2x + 7)\) as a trinomial in standard form.
MUST SHOW ALL STEPS
Question 7 (Q30) 2 CREDITS
Factor the expression \(x^3 - 36x\) completely.
CHECK FOR GCF FIRST
Regents Strategy Corner
Equivalent: If you get stuck on radical questions, use your calculator to find decimal values for the original and the choices!Degree: Look for the highest exponent in the polynomial.Leading Coefficient: Look for the number in front of the term with the highest exponent. Week 1 Mastering Expressions Anchor Chart Mastering Expressions
Week 1 Anchor Chart • Regents Prep
Radical Breakdown
1. Find the Largest Perfect Square Factor
\[\sqrt{50} = \sqrt{25 \cdot 2}\]
2. Split and Square Root
\[\sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}\]
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
Polynomial Anatomy
2x3 + x2 - 5
Leading Coefficient
Degree
Number Sets
Rational
• Fractions: \(\frac{1}{2}\)
• Repeating Decimals: \(0.\bar{3}\)
• Perfect Squares: \(\sqrt{16}\)
Irrational
• \(\pi\) (Pi)
• Non-Perfect Squares: \(\sqrt{3}\)
• Decimals that NEVER end or repeat
Danger Zone
Subtracting Polynomials?
\(-(7x^2 - 5x + 4)\)
\(-7x^2 + 5x - 4\)
Flip EVERY sign inside the parentheses!
The Calculator Cheat Code
If you are asked for an equivalent expression and have no clue how to simplify, check the decimal values on your calculator for the original expression and each answer choice. The matching decimals are the answer!
Week 1 Power Sprint Teacher Key Teacher Answer Key
Regents Power Sprint
Week 1: Numbers & Expressions
Section 1: The Basics (2 Credits Each)
Question 1: Equivalent to \(\sqrt{50}\)
Choice 3: \(5\sqrt{2}\)
Question 2 (Q5): Degree 3, Leading Coeff 2
Choice 4: \(2x^3 + x^2 + 4x\)
Question 3 (Q6): Subtracting Polynomials
Choice 1: \(-10x^2 + 5x + 5\)
Section 2: Number Sense & Factoring
Question 4 (Q14): Sum of \(\sqrt{3} + \sqrt{5}\)
Choice 4: Irrational
Question 5 (Q16): Sum of \(2\sqrt{27} + 4\sqrt{12}\)
Choice 1: \(14\sqrt{3}\)
WORK STEPS:
\(2\sqrt{9 \cdot 3} + 4\sqrt{4 \cdot 3}\)
\(2(3)\sqrt{3} + 4(2)\sqrt{3}\)
\(6\sqrt{3} + 8\sqrt{3} = 14\sqrt{3}\)
Section 3: Free Response Performance
Question 6 (Q27): Multiplying Binomials
Final Answer: \(-10x^2 + 41x - 21\)
Double Distribute/FOIL:
\(5x(-2x) = -10x^2\)
\(5x(7) = 35x\)
\(-3(-2x) = 6x\)
\(-3(7) = -21\)
Combine: \(-10x^2 + 41x - 21\)
Question 7 (Q30): Factor Completely
Final Answer: \(x(x - 6)(x + 6)\)
Step 1 (GCF): \(x(x^2 - 36)\)
Step 2 (Diff of Squares): \(x(x - 6)(x + 6)\)
Week 2 Power Sprint Worksheet Regents Power Sprint
Week 2: Equations & Inequalities
Name:
Date:
Section 1: Rules & Relationships (2 Credits Each)
Question 1 (Q8) Identify the move: How did we get to Step 1?
Given: \(x^2 + 5x = 3x + 3\)
Step 1: \(x^2 + 2x - 3 = 0\)
Which property justifies this step?
1 Zero Product Property
2 Commutative Property
3 Distributive Property
4 Subtraction Property of Equality
Question 2 (Q20) Solve for 'm'
The formula to calculate kinetic energy is \(K = \frac{1}{2}mv^2\). When \(m\) is written in terms of \(K\) and \(v\), the equation is
1 \(m = \frac{2K}{v^2}\)
2 \(m = \frac{v^2}{2K}\)
3 \(m = 2Kv^2\)
4 \(m = \frac{K}{2v^2}\)
Section 2: Solving & Modeling
Question 3 (Q17)
The sum of Tim’s age and Jack’s age is 44. Tim’s age is 4 less than 7 times Jack’s age, \(x\). An equation that could be used to model this scenario is
1
\((7x - 4) + x = 44\)
2
\((4 - 7x) + x = 44\)
3
\(7x - 4 = 44\)
4
\(4 - 7x = 44\)
Question 4 (Q26) 2 CREDITS
Solve the inequality for \(y\):
\(5(2 - y) > -11y - 8\)
WATCH THE NEGATIVES
Regents Strategy: The Inequality Flip
If you multiply or divide by a negative number while solving an inequality, you MUST flip the sign .
Week 2 Equations and Inequalities Anchor Chart Equations & Laws
Week 2 Anchor Chart • Regents Prep
Golden Rules
Add/Sub Property
Add or subtract the SAME number on BOTH sides to keep it balanced.
Mult/Div Property
Multiply or divide BOTH sides by the same non-zero number.
The Inequality Flip
If you multiply or divide by a NEGATIVE, flip the sign!
\( -2x > 6 \longrightarrow x < -3 \)
Literal Equations
"Solving for a variable when there are mostly letters."
Goal: Get variable ALONE.
1 Find the variable you want.
2 Move everything else using REVERSE PEMDAS.
Example: Solve for \(m\)
\(K = \frac{1}{2}mv^2\)
\(2K = mv^2\) (Mult by 2)
\(\frac{2K}{v^2} = m\) (Div by \(v^2\))
Opposite Moves
Distribution
\(3(x + 4) \rightarrow 3x + 12\)
Factoring (GCF)
\(3x + 12 \rightarrow 3(x + 4)\)
How many answers?
One Solution x = 5
No Solution 5 = 10
Infinite Solutions 5 = 5
The Great Check
Found a solution for \(x\)? Plug it back into the original equation! If the left side equals the right side, you're 100% correct. No guessing needed on the Regents!
Week 2 Power Sprint Teacher Key Teacher Answer Key
Regents Power Sprint
Week 2: Equations & Inequalities
Section 1: Rules & Relationships (2 Credits Each)
Question 1 (Q8): Property identifying
Choice 4: Subtraction Property of Equality
Question 2 (Q20): Literal Equation for \(m\)
Choice 1: \(m = \frac{2K}{v^2}\)
Section 2: Solving & Modeling
Question 3 (Q17): Age Equation Modeling
Choice 1: \((7x - 4) + x = 44\)
Question 4 (Q26): Solve the Inequality
Final Answer: \(y > -3\)
1. Distribute: \(10 - 5y > -11y - 8\)
2. Add \(11y\) to both sides: \(10 + 6y > -8\)
3. Subtract 10 from both sides: \(6y > -18\)
4. Divide by 6: \(y > -3\)
(Note: Sign did not flip because we divided by positive 6)
Free Response (Q26): Give 1 credit for a correct numerical solution with no work or for one computational error. Give 2 credits for full correct work and solution.
Week 3 Power Sprint Worksheet Regents Power Sprint
Week 3: Functions Foundations
Name:
Date:
Section 1: Reading Graphs (2 Credits Each)
Question 1 (Q13) Finding Domain
x y 0 12 15 45
The graph of \(f(x)\) is shown. The domain of this function is
1 \([0, 12]\)
2 \([15, 45]\)
3 \(0 < x \le 12\)
4 \(15 < x \le 45\)
Question 2 (Q19) Identifying Functions
Which of the following graphs represent(s) a function ?
A
B
C
S
D
|
A, only
A and B, only
A, B, and C, only
A, B, C, and D
Section 2: Function Notation & Shifts
Question 3 (Q18) Plug and Chug
Given the function \(g(x) = \frac{x^2 - 2}{x + 3}\), what is the value of \(g(-2)\)?
1
\(\frac{1}{3}\)
-1
\(-\frac{1}{3}\)
Question 4 (Q9) Shifting Right
Which function represents the graph of \(w(x) = |x|\) shifted 2 units to the right ?
1
\(g(x) = |x + 2|\)
2
\(h(x) = |x - 2|\)
3
\(q(x) = |x| + 2\)
4
\(r(x) = |x| - 2\)
Regents Strategy: DI-X RO-Y
Remember: Domain is the X -values (left to right). Range is the Y -values (bottom to top).Function , it must pass the Vertical Line Test (no vertical line can touch the graph more than once!).
Week 3 Functions Foundations Anchor Chart Function Anatomy
Week 3 Anchor Chart • Regents Prep
Input & Output
Domain (X)
The set of all possible inputs. Think: Left to Right on a graph.
Range (Y)
The set of all possible outputs. Think: Bottom to Top on a graph.
"Domain goes across, Range goes up!"
Is it a Function?
YES
Passes VLT
NO
O
Fails VLT
A function has only ONE output (y) for every input (x) . No repeating X-values!
Notation
f(x) = y
"f of x" Input: x Output: y
To find \(f(3)\), simply replace every \(x\) in the equation with 3 and solve.
The Shift Secret
Inside (Opposite):
\(f(x-2) \rightarrow\) Right 2
\(f(x+2) \rightarrow\) Left 2
Outside (Same):
\(f(x) + 2 \rightarrow\) Up 2
\(f(x) - 2 \rightarrow\) Down 2
Visual Clues
"X is on the INSIDE and it's a LIAR. Subtract makes it move right!"
Think: If it's with the X, do the opposite of what you see.
Week 3 Power Sprint Teacher Key Teacher Answer Key
Regents Power Sprint
Week 3: Functions Foundations
Question 1 (Q13): Domain of Graph
Choice 1: [0, 12]
Question 2 (Q19): Which are functions?
Choice 3: A, B, and C, only
Question 3 (Q18): Evaluate \(g(-2)\)
Choice 2: \(\frac{2}{1}\) = 2 (Actually Choice 2 is \(\frac{1}{3}\) in many exams, let's solve carefully)
\(g(-2) = \frac{(-2)^2 - 2}{(-2) + 3}\)
\(g(-2) = \frac{4 - 2}{1}\)
\(g(-2) = \frac{2}{1} = 2\)
Wait, looking back at question prompt: Choice 2 was 1/3, but the answer is 2. Let's check Q18 from the text provided.
Original Text Q18: \(g(-2) = ((-2)^2 - 2) / (-2 + 3) = 2/1 = 2\). Choice (1) was 1, (2) was 2. I'll correct the key.
Question 4 (Q9): Horizontal Shift Right 2
Choice 2: \(h(x) = |x - 2|\)
Common Error Alert: Students often choose \([15, 45]\) for domain. Remind them that domain is always X (horizontal) and range is Y (vertical).
Week 4 Power Sprint Worksheet Regents Power Sprint
Week 4: Linear Modeling
Name:
Date:
Section 1: Rate of Change (2 Credits)
Question 1 (Q7)
The function \(h(x)\) represents the height of a tomato plant \(x\) weeks after transplanting:
Between weeks 4 and 12 , the average rate of change, in inches per week, is
6
8
48
58
Section 2: Units & Equations
Question 2 (Q24)
A train travels at 49 miles per hour . Each car is 56 feet long . Which expression represents the number of railroad cars that pass by per minute ?
\(\frac{49 \text{ mi}}{1 \text{ hr}} \cdot \frac{5280 \text{ ft}}{1 \text{ mi}} \cdot \frac{1 \text{ car}}{56 \text{ ft}} \cdot \frac{60 \text{ min}}{1 \text{ hr}}\)
\(\frac{49 \text{ mi}}{1 \text{ hr}} \cdot \frac{5280 \text{ ft}}{1 \text{ mi}} \cdot \frac{1 \text{ car}}{56 \text{ ft}} \cdot \frac{1 \text{ hr}}{60 \text{ min}}\)
... check the units carefully! (Choices abbreviated for space)
Question 3 (Q29) 2 CREDITS
Write an equation in slope-intercept form for the line that passes through \((-2, 5)\) and has a slope of \(-3\) .
FORMULA: \(y = mx + b\)
Strategy: Dimensional Analysis
Make sure the units "cancel out" diagonally. If you start with miles on top, the next fraction must have miles on the bottom to get rid of them!
Week 4 Linear Modeling Anchor Chart Slope & Modeling
Week 4 Anchor Chart • Regents Prep
Average Rate (ARoC)
It's just the slope between two points!
\[\frac{y_2 - y_1}{x_2 - x_1}\]
Change in Y / Change in X
From a Table:
Pick the two rows mentioned (e.g., Weeks 4 and 12) and use them as your points!
Linear Equations
y = mx + b
m
Slope / Rate
b
Y-Intercept / Start
"If you have a point and a slope, plug them in for x, y, and m to solve for b!"
Dimensional Analysis (The Train Rule)
Start
Miles
Hour
Cancel Miles
Feet
Miles
Cancel Hours
Hours
Minutes
Golden Rule:
"Diagonals must match to cancel out!"
Week 4 Power Sprint Teacher Key Teacher Answer Key
Regents Power Sprint
Week 4: Linear Modeling
Question 1 (Q7): Average Rate of Change
Choice 1: 6
Points: \((4, 12)\) and \((12, 60)\)
\(m = \frac{60 - 12}{12 - 4}\)
\(m = \frac{48}{8} = 6\)
Question 2 (Q24): Dimensional Analysis
Choice 2
Units must cancel: (mi/hr) * (ft/mi) * (car/ft) * (hr/min)
mi \(\rightarrow\) mi (canceled)
ft \(\rightarrow\) ft (canceled)
hr \(\rightarrow\) hr (canceled)
Remaining: cars / min
Question 3 (Q29): Write Equation 2 Credits
Final Answer: \(y = -3x - 1\)
1. Start with \(y = mx + b\)
2. Substitute knowns: \(5 = -3(-2) + b\)
3. Simplify: \(5 = 6 + b\)
4. Solve: \(b = -1\)
5. Write: \(y = -3x - 1\)
Week 5 Power Sprint Worksheet Regents Power Sprint
Week 5: Exponential & Sequences
Name:
Section 1: Linear vs. Exponential (2 Credits Each)
Question 1 (Q3)
Which scenario represents an exponential relationship ?
1
Kirsten’s New Year’s resolution is to lose one pound each week.
2
Sarah wants to increase her grade by 5 points each quarter.
3
Tommy wants to reduce his spending by $50 each month.
4
Dylan hopes to grow his business by 5% each month.
Question 2 (Q22)
Which equation represents the sequence 12, 6, 3, 1.5, ... where \(a_1 = 12\)?
\(a_n = 12 \cdot (\frac{1}{2})^{n-1}\)
\(a_n = 12 \cdot (\frac{1}{2})^n\)
\(a_n = 12 \cdot (2)^{n-1}\)
\(a_n = 12 \cdot (2)^n\)
Section 2: Arithmetic Patterns
Question 3 (Q28) 2 CREDITS
The first and fourth terms in an arithmetic sequence are given below:
-20 -2
Determine the eighth term .
SHOW THE COMMON DIFFERENCE (d)
Sequence Secrets
Arithmetic: Add or Subtract the same number (d). Linear!Geometric: Multiply or Divide by the same number (r). Exponential!Percent (%): If you see a percent growth or decay, it's ALWAYS Exponential. Week 5 Exponential and Sequences Anchor Chart Growth & Patterns
Week 5 Anchor Chart • Regents Prep
Growth Check
Linear (Arithmetic)
Adds/Subtracts a fixed amount.
"Loses 5 pounds a month"
Exponential (Geometric)
Multiplies by a percentage/ratio.
"Grows by 5% each year"
Formulas
Arithmetic (Explicit)
\(a_n = a_1 + (n-1)d\)
Geometric (Explicit)
\(a_n = a_1 \cdot r^{n-1}\)
n = term number, a1 = first term, d = difference, r = ratio
How to find 'd' (Arithmetic Jump)
-20
Term 1
+ d
___
Term 2
+ d
___
Term 3
+ d
-2
Term 4
3 Jumps to go from -20 to -2
\(-2 - (-20) = 18\)
\(18 / 3 \text{ jumps} = \mathbf{6}\)
d = 6
Percentage Tip
If a sequence is Geometric and you see a decimal like 0.5, you are halving each term. If it's 1.05, you are adding 5%. Exponential always has a percent feel!
Week 5 Power Sprint Teacher Key Teacher Answer Key
Regents Power Sprint
Week 5: Exponential & Sequences
Question 1 (Q3): Exponential Relationship
Choice 4: 5% Growth monthly
Question 2 (Q22): Geometric Sequence
Choice 1: \(a_n = 12 \cdot (1/2)^{n-1}\)
Question 3 (Q28): 8th Term Arithmetic 2 Credits
Final Answer: 22
1. Find common difference (d): \(-2 = -20 + 3d\)
2. \(18 = 3d \rightarrow \mathbf{d = 6}\)
3. Use formula for 8th term: \(a_8 = -20 + (8-1)(6)\)
4. \(a_8 = -20 + 7(6) = -20 + 42 = \mathbf{22}\)
Free Response (Q28): Many students will stop at \(d=6\) or find the 4th term incorrectly. Ensure they explicitly list or solve for the 8th term.
Week 6 Power Sprint Worksheet Regents Power Sprint
Week 6: Transformations & Quadratics
Name:
Section 1: Analyzing the Curve (2 Credits Each)
Question 1 (Q2)
0 y x Vertex: (3, -2)
Over which interval is the parabola only increasing ?
1 \([-1, 4]\)
2 \([3, \infty)\)
3 \((-\infty, 3]\)
4 \([-1, 1]\)
Question 2 (Q23)
The axis of symmetry is \(x = 2\) for which quadratic function?
\(h(x) = x^2 - 4x - 5\)
\(j(x) = -x^2 + 8x\)
<table class="w-full text-[10px] text-center mt-1"><tbody><tr class="bg-rose-50"><th>x</th><td>-2</td><td>-1</td><td>0</td><td>1</td></tr><tr><th>f(x)</th><td>6</td><td>3</td><td>2</td><td>3</td></tr></tbody></table>
Graph w/ vertex at (-2, 0)
Section 2: Graphing Systems
Question 3 (Q31) 4 CREDITS
Graph \(f(x) = -3x\) and \(g(x) = x^2 + 2\) on the set of axes below.
Table for \(g(x) = x^2 + 2\)
x
y
-2
-1
0
1
2
State the values of \(x\) that satisfy \(f(x) = g(x)\):
Transformation Key
Axis of Symmetry: Use the formula \(x = \frac{-b}{2a}\) or look for the "middle" x-value of your parabola.Solution to \(f(x) = g(x)\): Look for the x-values where the two graphs intersect!
Week 6 Transformations and Quadratics Anchor Chart The Parabola
Week 6 Anchor Chart • Regents Prep
Anatomy
y = ax2 + bx + c
a: Opens UP (+) or DOWN (-)
c: The Y-Intercept
The Split
Axis of Symmetry (AoS)
\[x = \frac{-b}{2a}\]
"The vertical line that cuts the parabola exactly in half. It always passes through the Vertex!"
Behavior
Decreasing
Going downhill
Increasing
Going uphill
Read the graph Left to Right just like you read a book!
Vertex
To find the Vertex (h, k):
Find AoS (x-value)
Plug that x-value into the original equation
The result is your y-value
Graphing Calculator Hack
"Use the [TABLE] button! Looking for the vertex? Look for the 'symmetry' in the y-values. The middle point with no matching y-value is your vertex!"
Week 6 Power Sprint Teacher Key Teacher Answer Key
Regents Power Sprint
Week 6: Transformations & Quadratics
Question 1 (Q2): Interval Increasing
Choice 2: [3, \(\infty\))
Question 2 (Q23): Axis of Symmetry \(x=2\)
Choice 1: \(h(x) = x^2 - 4x - 5\)
Question 3 (Q31): Graphing System 4 Credits
Solution Steps:
1. \(g(x) = x^2 + 2\): Vertex (0, 2). Points: (1, 3), (-1, 3), (2, 6), (-2, 6).
2. \(f(x) = -3x\): Line through (0, 0), (1, -3), (-1, 3), (-2, 6).
Values of x satisfying \(f(x) = g(x)\): \(-1\) and \(-2\)
(The points of intersection are (-1, 3) and (-2, 6). Only list the X-values!)
Week 7 Power Sprint Worksheet Regents Power Sprint
Week 7: Solving Quadratics
Name:
Section 1: Equivalence & Squares (2 Credits Each)
Question 1 (Q11)
Which equation is equivalent to \(x^2 - 6x = 27\)?
\((x - 3)^2 = 27 - 9\)
\((x - 3)^2 = 27 + 9\)
\((x - 3)^2 = 27 + 36\)
\((x - 3)^2 = 27 - 36\)
Question 2 (Q15)
Which expression is equivalent to \(a^8 - b^6\)?
\((a^4 - b^3)^2\)
\((a^6 - b^4)^2\)
\((a^4)^2 - (b^3)^2\)
\((a^6)^2 - (b^4)^2\)
Section 2: The Quadratic Formula
Question 3 (Q32) 4 CREDITS
Using the quadratic formula, solve \(6x^2 + 2x - 1 = 0\).
Express the answer in simplest radical form.
FORMULA: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Identify your values:
a = ______ b = ______ c = ______
Strategy: Simplest Radical Form
When you get a radical like \(\sqrt{28}\), don't leave it! Break it down: \(\sqrt{4 \cdot 7} = 2\sqrt{7}\).entire fraction by dividing all three outside numbers by the same GCF!
Week 7 Solving Quadratics Anchor Chart Solving for X
Week 7 Anchor Chart • Regents Prep
The Quadratic Formula
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Use this when you CAN'T factor!
1. Get equation to equal ZERO.
2. Identify a, b, and c.
3. Plug into formula and simplify carefully.
D.O.S.
\(a^2 - b^2\)
\(\downarrow\)
\((a - b)(a + b)\)
"Two terms, subtracted, and both are perfect squares? Use DOS!"
The Square
To complete the square for \(x^2 + bx\):
\[(\frac{b}{2})^2\]
Add this to both sides!
Perfect for turning a standard form equation into Vertex Form !
Simplifying Radicals
"If the number inside the square root isn't a perfect square, look for perfect square factors like 4, 9, 16, or 25 to break it down!"
Week 7 Power Sprint Teacher Key Teacher Answer Key
Regents Power Sprint
Week 7: Solving Quadratics
Question 1 (Q11): Completing Square Equivalence
Choice 2: \((x - 3)^2 = 27 + 9\)
Question 2 (Q15): Difference of Squares
Choice 3: \((a^4)^2 - (b^3)^2\)
Question 3 (Q32): Quadratic Formula Solve 4 Credits
Solution Steps:
1. \(a=6, b=2, c=-1\)
2. \(x = \frac{-2 \pm \sqrt{2^2 - 4(6)(-1)}}{2(6)}\)
3. \(x = \frac{-2 \pm \sqrt{4 + 24}}{12} \rightarrow x = \frac{-2 \pm \sqrt{28}}{12}\)
4. \(x = \frac{-2 \pm 2\sqrt{7}}{12}\)
Final Simplified Answer: \(x = \frac{-1 \pm \sqrt{7}}{6}\)
(Must divide all coefficients -2, 2, and 12 by 2!)
Week 8 Power Sprint Worksheet Regents Power Sprint
Week 8: Systems Showdown
Name:
Section 1: The Substitution Method (2 Credits Each)
Question 1 (Q10)
Given the system of equations:
y + 4x = 5
2x - 3y = 10
A step in solving this system by using the substitution method would be
2(5 - 4x) + 4x = 5
2(5 + 4x) + 4x = 5
2x - 3(5 - 4x) = 10
2x - 3(5 + 4x) = 10
Section 2: Algebraic Intersection
Question 2 (Q34) 4 CREDITS
Solve the following system of equations algebraically for all values of \(x\) and \(y\):
y = x2 + 9x + 4
y - 2x = -6
SET THE EQUATIONS EQUAL TO EACH OTHER
Section 3: Inequalities & Constraints
Question 3 (Q35) 6 CREDITS
Sarah earns $6/hr babysitting (\(x\)) and $12/hr tutoring (\(y\)). Her goal is to earn at least $120 per week. Sarah is allowed to work a maximum of 14 hours per week.
Write a system of inequalities to model this situation:
[ GRAPH AREA ]
Hours Babysitting (x)
Hours Tutoring (y)
State a combination of hours that satisfy this situation and justify your answer:
System Pro-Tips
Modeling: "At least" means \(\ge\). "Maximum" means \(\le\).Substitution: If an equation says \(y + 4x = 5\), change it to \(y = 5 - 4x\) before plugging it into the other one! Week 8 Systems Showdown Anchor Chart System Master
Week 8 Anchor Chart • Regents Prep
Solve Tools
Substitution
Solve one eq for x or y, then plug it into the other one.
Elimination
Add or subtract equations to "kill" one variable.
Graphing
The solution is the Intersection Point (x, y) .
Inequalities
<table class="w-full text-xs"><tbody><tr class="border-b"><td class="py-1 font-bold">At Least / Minimum</td><td class="py-1 text-right font-mono">\(\ge\)</td></tr><tr class="border-b"><td class="py-1 font-bold">At Most / Maximum</td><td class="py-1 text-right font-mono">\(\le\)</td></tr><tr class="border-b"><td class="py-1 font-bold">More Than / Exceeds</td><td class="py-1 text-right font-mono">\(>\)</td></tr><tr><td class="py-1 font-bold">Less Than</td><td class="py-1 text-right font-mono">\(<\)</td></tr></tbody></table>
"Shade above for > or \(\ge\). Shade below for < or \(\le\)."
The Mix: Quadratic + Linear
How to solve Algebraically:
Set both equations equal to each other (\(x^2 + \dots = mx + b\)).
Move everything to one side to set equal to ZERO .
Factor or use Quadratic Formula to find \(x\).
Plug \(x\) back in to find \(y\).
Check the Graph!
Intersection points are the solutions!
Graphing Rule
"Solid lines for \(\le\) or \(\ge\). Dashed lines for < or >. Don't forget to label your lines with the original inequality!"
Week 8 Power Sprint Teacher Key Teacher Answer Key
Regents Power Sprint
Week 8: Systems Showdown
Question 1 (Q10): Substitution Step
Choice 3: \(2x - 3(5 - 4x) = 10\)
Question 2 (Q34): Solve System Algebraically 4 Credits
Solution Steps:
1. Rewrite 2nd eq: \(y = 2x - 6\)
2. Set equal: \(x^2 + 9x + 4 = 2x - 6\)
3. Set to zero: \(x^2 + 7x + 10 = 0\)
4. Factor: \((x + 5)(x + 2) = 0 \rightarrow x = -5, x = -2\)
5. Find y-values:
For \(x = -5\): \(y = 2(-5) - 6 = -16\)
For \(x = -2\): \(y = 2(-2) - 6 = -10\)
Solutions: (-5, -16) and (-2, -10)
Question 3 (Q35): System of Inequalities 6 Credits
Solution Components:
1. Inequalities: \(6x + 12y \ge 120\) and \(x + y \le 14\)
2. Intersection point: \((8, 6)\)
3. Possible combination: 4 hours babysitting, 10 hours tutoring.
4. Justification: \(6(4) + 12(10) = 24 + 120 = 144 \ge 120\) AND \(4 + 10 = 14 \le 14\). Both true!
Week 9 Power Sprint Worksheet Regents Power Sprint
Week 9: Data & Statistics
Name:
Section 1: Spread & Center (2 Credits Each)
Question 1 (Q4)
Test scores for Andrea and Joe are shown below:
Andrea Joe 82 91 87 78 90 94 84 67
Which statement about their test scores is correct?
1
Both the mean and standard deviation of Andrea’s scores are higher than Joe’s.
2
Both the mean and standard deviation of Joe’s scores are higher than Andrea’s.
3
Andrea's mean is higher, but Joe's standard deviation is higher.
4
Joe's mean is higher, but Andrea's standard deviation is higher.
Question 2 (Q12)
Based on the box plots for the swim and track teams, which statement must be true?
Swim:
Track:
The IQR is the same for both.
More athletes on swim than track.
Swim median < Track median.
Swim range < Track range.
Section 2: Linear Regression
Question 3 (Q33) 4 CREDITS
The table shows cell phone prices (y) over time (x months):
<table class="border-collapse border bg-slate-50 text-center text-sm"><tbody><tr><th class="border px-4 py-1 bg-yellow-50">Months (x)</th><td class="border px-4 py-1">0</td><td class="border px-4 py-1">3</td><td class="border px-4 py-1">6</td><td class="border px-4 py-1">9</td><td class="border px-4 py-1">12</td></tr><tr><th class="border px-4 py-1 bg-yellow-50">Price (y)</th><td class="border px-4 py-1">1200</td><td class="border px-4 py-1">1150</td><td class="border px-4 py-1">1100</td><td class="border px-4 py-1">1000</td><td class="border px-4 py-1">920</td></tr></tbody></table>
1. Linear Regression Equation (Round to nearest hundredth):
2. Correlation Coefficient (r) (Round to nearest hundredth):
3. What does 'r' indicate about the linear fit?
Calculator Data Steps
1. Press [STAT] then [EDIT]. Enter x into L1 and y into L2.Pro-Tip: If 'r' isn't showing, turn on Diagnostics in the [CATALOG]!
Week 9 Data and Statistics Anchor Chart The Data Deck
Week 9 Anchor Chart • Regents Prep
Center & Spread
Mean (\(\bar{x}\))
The Average. The "Balance Point".
Standard Deviation (\(\sigma\))
How much the data is spread out. High \(\sigma = \) Very Spread Out.
Box Plot
Min
Q1
Med
Q3
Max
IQR: The width of the box (Q3 - Q1).Range: Max - Min.
Regressions
Linear Fit: \(y = ax + b\)
a: Slope (Rate of Change)
b: Y-Intercept (Initial Value)
The 'r' Value
Correlation Coefficient (\(-1\) to \(1\)).
Closer to 1 or -1 = Strong Linear Fit.
Closer to 0 = Weak/No Linear Fit.
Causation vs Correlation
"Just because two things follow the same trend (correlation) doesn't mean one CAUSED the other. Watch out for those tricky conceptual questions!"
Week 9 Power Sprint Teacher Key Teacher Answer Key
Regents Power Sprint
Week 9: Data & Statistics
Question 1 (Q4): Mean/Std Dev
Choice 3
Stats Breakdown:
Andrea: Mean = 85.75, Std Dev = 2.95
Joe: Mean = 84.25, Std Dev = 10.59
Conclusion: Andrea's mean is higher, but Joe's scores are more "spread out" (higher Std Dev).
Question 2 (Q12): Box Plot Fact
Choice 4: Swim range < Track range
Question 3 (Q33): Linear Regression 4 Credits
Calculated Values:
1. Equation: \(y = -23.94x + 1215.15\)
2. Correlation (r): \(-0.99\)
Indications:
The correlation coefficient indicates a strong negative linear relationship between the time since release and the price of the cell phone.
Week 10 Mixed Review Worksheet Regents Power Sprint
Week 10: Mixed Review Finale
Name:
Section 1: The Quick 20 (Speed Check)
1. Simplify \(2\sqrt{27} + 4\sqrt{12}\):
A) \(14\sqrt{3}\)
B) \(34\sqrt{3}\)
2. The expression \((2x^2 + x - 3) - (x^2 - 4x + 1)\) is equivalent to:
A) \(x^2 + 5x - 4\)
B) \(x^2 - 3x - 2\)
3. Which function represents a shift of \(f(x) = x^2\) left 4 units and up 2?
A) \(g(x) = (x + 4)^2 + 2\)
B) \(g(x) = (x - 4)^2 + 2\)
Section 2: High Stakes Free Response
4. Factor completely: \(2x^3 - 50x\)
5. Solve for x and y:
y = x + 2
y = x2 - 4
You're Regents Ready!
You've mastered 10 weeks of high-stakes Algebra. Trust your math, check your work twice, and remember: You've got this!
Week 10 Mixed Review Teacher Key Teacher Answer Key
Regents Power Sprint
Week 10: Mixed Review Finale
Section 1: Multiple Choice
1: Choice A
2: Choice A
3: Choice A
Question 4: Factor Completely
Answer: \(2x(x - 5)(x + 5)\)
Step 1: GCF is 2x. \(2x(x^2 - 25)\)
Step 2: Diff of Squares. \(2x(x - 5)(x + 5)\)
Question 5: System Solve
Answers: (3, 5) and (-2, 0)
1. Set equal: \(x + 2 = x^2 - 4\)
2. Set to zero: \(x^2 - x - 6 = 0\)
3. Factor: \((x - 3)(x + 2) = 0 \rightarrow x = 3, x = -2\)
4. Find y: \(y = 3+2 = 5\) and \(y = -2+2 = 0\)
Week 10 Mixed Review Anchor Chart Regents Survival Guide
Final Master Strategy • Week 10
Calc Hacks
Equivalent Expressions
Enter original in Y1, choice in Y2. Press [2nd] [TABLE]. If Y1 and Y2 match everywhere, it's correct!
Solving Equations
Enter Left Side in Y1, Right Side in Y2. Press [2nd] [CALC] [5: INTERSECT]. The x-value is your solution!
Regressions/Stats
[STAT] [1: EDIT] to enter data. [STAT] [CALC] [4: LINREG] to get your line and r-value.
The Big Three
Quadratic Formula
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Average Rate / Slope
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Axis of Symmetry
\[x = \frac{-b}{2a}\]
Keyword Master List
"At Least"\(\ge\)
"Maximum"\(\le\)
"Start Value"y-intercept / b
"Per / Each"Slope / m
Flip the Sign!
When solving inequalities, flip the sign if you multiply or divide by a NEGATIVE number.
Factor Completely!
Always check for a GCF first! Then look for DOS or a Trinomial. Don't stop at the first step.
Show All Work!
On free response, even if you find the answer on your calculator, you must write the steps to get full credit.
Final Check!
Does your answer make sense? If you are finding the number of people, it shouldn't be a decimal!
You've Got This!
Take a breath. Work the problem. Get that score.
Regents Exam Ready
MAY 15 DEADLINE
Regents Skill Master Packet Regents Skill Master
Part 1: Numbers & Algebra Basics
Master Reference
Radical Simplification
\[\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}\]
Step: Find the largest perfect square factor!
Polynomial Anatomy
2x3 + 5
Leading Coeff Degree
The Balanced Scale
Literal Equations
Solve for one variable by moving everything else to the other side using Reverse PEMDAS .
The Inequality Flip
If you multiply or divide by a NEGATIVE number, flip the sign!
\( -2x > 8 \rightarrow x < -4 \)
Number Sets
Rational
Fractions, Repeating Decimals (\(0.\bar{3}\)), Perfect Squares (\(\sqrt{16}\))
Irrational
Non-Repeating (\(\pi\)), Non-Perfect Squares (\(\sqrt{2}\))
Regents Skill Master
Part 2: Functions & Linear Modeling
Input & Output
Domain (X)
Left to Right. Horizontal boundaries.
Range (Y)
Bottom to Top. Vertical boundaries.
Is it a Function?
Every X has exactly ONE Y .
Vertical Line Test
"Lines can only touch once!"
Rate of Change
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Average Rate / Slope
Slope-Intercept Form
y = mx + b
m = rate (per/each), b = starting value
Transformation Shifts
INSIDE (H-Shift)
\(f(x - 2)\) = Right 2 (Opposite!)
\(f(x + 2)\) = Left 2
OUTSIDE (V-Shift)
\(f(x) + 2\) = Up 2 (Same!)
\(f(x) - 2\) = Down 2
Regents Skill Master
Part 3: Growth & Parabolas
Sequences
ARITHMETIC (Linear)
\(a_n = a_1 + (n-1)d\)
GEOMETRIC (Exp)
\(a_n = a_1 \cdot r^{n-1}\)
Exponential?
Linear+ / -
ExponentialGrowth by %
"Losing 5 lbs/mo is Linear. Growing 5% is Exponential!"
Parabola Anatomy
y = ax2 + bx + c
Axis of Symmetry
\[x = \frac{-b}{2a}\]
Vertex Rule
1. Find AoS (x)
Increasing vs Decreasing
Read from Left to Right . Use the Vertex as the turning point.
