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Monday — Domain & Range Rebuild
Tuesday — Factoring Patterns
Wednesday — Polynomial Arithmetic + Linear Equations/Inequalities
Thursday — Mixed Part-I Timed Set
Friday — Re-do Q13 and Q30, plus Sibling Problems
Saturday — Full Part I + Part II Timed

Week 1 — Pattern Recognition + Quick Wins

Goal this week: lock in the easy point-recovery items — domain/range, factoring patterns, and the universal heuristics. Build the Mistake-Log habit.

Time: ~60 min weekdays, ~75 min Saturday. Sunday off.

Heuristics to apply daily: H1 (read prompt twice), H2 (name the structure), H4 (simplest form), H5 (re-do heavy arithmetic).

How to use the practice problems: every problem has a Hint and a Show answer drop-down. Attempt the problem on paper first. If you’re stuck, peek at the hint. Only reveal the answer after you’ve committed to one of your own. Every miss gets a Mistake Log entry tagged with the H-bucket that would have caught it.

Monday — Domain & Range Rebuild

Objective

Burn into muscle memory: domain lives on the $x$ -axis, range lives on the $y$ -axis. This single error cost a point on Q13.

Instructions

Set up your binder with a fresh Mistake Log notebook.
For each of the 10 graph descriptions below, sketch the graph on graph paper. Use green pen for the $x$ -axis extent and red pen for the $y$ -axis extent.
Write the domain and range underneath, using interval notation.
After each, open Hint only if stuck, then check against the answer. For every miss, write a Mistake Log entry tagged “H2 — domain/range swap.”

Practice problems

For each function $f(x)$ , state the domain and range.

1. Line from $(-3, -2)$ to $(5, 6)$ , both endpoints closed.

Hint

Closed endpoints use $\leq$ . Domain runs between the $x$ -coordinates.

Show answer

Domain: $-3 \leq x \leq 5$ . Range: $-2 \leq y \leq 6$ .

2. Line from open $(0, 1)$ to closed $(10, 21)$ .

Hint

Open dot → strict $<$ ; closed dot → $\leq$ .

Show answer

Domain: $0 < x \leq 10$ . Range: $1 < y \leq 21$ .

3. Parabola $y = x^2$ , all real $x$ .

Hint

Any real number squared is $\geq 0$ .

Show answer

Domain: all reals. Range: $y \geq 0$ .

4. Parabola $y = -x^2 + 4$ .

Hint

The negative leading coefficient flips the parabola; its maximum is at the vertex.

Show answer

Domain: all reals. Range: $y \leq 4$ .

5. Open dot at $(-4, 2)$ , closed dot at $(8, 11)$ , line between.

Hint

Open on the left → strict $<$ ; closed on the right → $\leq$ .

Show answer

Domain: $-4 < x \leq 8$ . Range: $2 < y \leq 11$ .

6. $y = \sqrt{x}$ .

Hint

You can’t take a real square root of a negative number. Output is never negative.

Show answer

Domain: $x \geq 0$ . Range: $y \geq 0$ .

7. $y = |x|$ .

Hint

Absolute value is defined for every real number; outputs are never negative.

Show answer

Domain: all reals. Range: $y \geq 0$ .

8. Horizontal line $y = 5$ for $-2 \leq x \leq 2$ .

Hint

A horizontal segment has only one $y$ -value.

Show answer

Domain: $-2 \leq x \leq 2$ . Range: $\{5\}$ .

9. $y = 1/x$ for $x > 0$ .

Hint

The reciprocal of a positive number is positive. As $x \to 0^+$ , $y \to \infty$ ; as $x \to \infty$ , $y \to 0^+$ .

Show answer

Domain: $x > 0$ . Range: $y > 0$ .

10. Piecewise: closed at $(-5, -3)$ rising to open at $(0, 7)$ , then closed at $(0, 4)$ rising to closed at $(6, 10)$ .

Hint

Combine the $x$ -intervals for the domain. For the range, union the two $y$ -intervals.

Show answer

Domain: $-5 \leq x \leq 6$ . Range: $-3 \leq y \leq 10$ (combined).

Self-check question

If a graph has an open dot, is that endpoint included in the domain/range? (No — use $<$ or $>$ , not $\leq$ or $\geq$ .)

Tuesday — Factoring Patterns

Objective

Recognize the four core factoring patterns within 5 seconds of seeing them. Practice the H2 heuristic: name the structure aloud before you write anything.

Instructions

For each expression, say aloud which of the four patterns it matches: GCF, Difference of Squares (DOS), Perfect-Square Trinomial (PST), or Standard Trinomial (ST). Then factor it completely. Use the hint only if you can’t see the pattern; reveal the answer only after committing to one of your own.

Practice problems

1. $x^2 - 49$

Hint

Two terms, both perfect squares, minus sign.

Show answer

Pattern: DOS. Factored: $(x - 7)(x + 7)$ .

2. $x^2 + 10x + 25$

Hint

$25 = 5^2$ and $10 = 2 \cdot 5$ .

Show answer

Pattern: PST. Factored: $(x + 5)^2$ .

3. $x^2 + 7x + 12$

Hint

Find two integers that multiply to $12$ and add to $7$ .

Show answer

Pattern: ST. Factored: $(x + 3)(x + 4)$ .

4. $4x^3 + 12x^2$

Hint

Every term shares $4x^2$ .

Show answer

Pattern: GCF. Factored: $4x^2(x + 3)$ .

5. $x^3 - 36x$

Hint

Pull $x$ first, then look at what’s left.

Show answer

Pattern: GCF + DOS. $x(x^2 - 36) = x(x - 6)(x + 6)$ .

6. $9x^2 - 16$

Hint

$(3x)^2 - 4^2$ .

Show answer

Pattern: DOS. Factored: $(3x - 4)(3x + 4)$ .

7. $x^2 - 14x + 49$

Hint

$49 = 7^2$ and $14 = 2 \cdot 7$ .

Show answer

Pattern: PST. Factored: $(x - 7)^2$ .

8. $x^2 + 5x - 14$

Hint

Two integers that multiply to $-14$ and add to $+5$ .

Show answer

Pattern: ST. Factored: $(x + 7)(x - 2)$ .

9. $2x^4 - 32$

Hint

GCF first, then DOS — and look again at what’s inside.

Show answer

Pattern: GCF + DOS (twice). $2(x^4 - 16) = 2(x^2 - 4)(x^2 + 4) = 2(x - 2)(x + 2)(x^2 + 4)$ .

10. $50x^3 - 18x$

Hint

Pull $2x$ first, then DOS on $25x^2 - 9$ .

Show answer

Pattern: GCF + DOS. $2x(25x^2 - 9) = 2x(5x - 3)(5x + 3)$ .

11. $x^2 - 6x + 9$

Hint

$9 = 3^2$ and $6 = 2 \cdot 3$ .

Show answer

Pattern: PST. Factored: $(x - 3)^2$ .

12. $x^2 - 64$

Hint

Two perfect-square terms, subtracted.

Show answer

Pattern: DOS. Factored: $(x - 8)(x + 8)$ .

13. $6x^2 + 9x$

Hint

$3x$ divides both terms.

Show answer

Pattern: GCF. Factored: $3x(2x + 3)$ .

14. $x^2 + x - 12$

Hint

Two integers that multiply to $-12$ and add to $+1$ .

Show answer

Pattern: ST. Factored: $(x + 4)(x - 3)$ .

15. $4x^2 - 4x + 1$

Hint

$(2x)^2$ and $1^2$ , with middle term $= 2(2x)(1)$ .

Show answer

Pattern: PST. Factored: $(2x - 1)^2$ .

Self-check question

What’s the giveaway that distinguishes $x^2 - 36$ (DOS) from $x^2 - 12x + 36$ (PST)? (Two terms = DOS; three terms with the middle = $\pm 2ab$ pattern = PST.)

Wednesday — Polynomial Arithmetic + Linear Equations/Inequalities

Objective

Tighten up the basics: sign distribution in polynomial subtraction, the box method for products, and the inequality-flip rule.

Instructions

Do all 14 problems. Show full work. Use the box method for every multiplication, even simple ones — building the habit matters more than speed this week.

Part A — Polynomial arithmetic

1. $(3x^2 - 5x + 7) - (x^2 + 2x - 4)$

Hint

Distribute the minus across all three terms of the second polynomial, then combine like terms.

Show answer

$2x^2 - 7x + 11$ .

2. $(-2x^2 + 6) - (4x^2 - 3x + 1)$

Hint

After distributing the minus: $-2x^2 + 6 - 4x^2 + 3x - 1$ .

Show answer

$-6x^2 + 3x + 5$ .

3. $(2x + 3)(x - 5)$

Hint

Box method or FOIL. Middle term is $-10x + 3x$ .

Show answer

$2x^2 - 7x - 15$ .

4. $(3x - 4)(2x + 1)$

Hint

Watch the signs on the $-4$ row of the box.

Show answer

$6x^2 - 5x - 4$ .

5. $(x + 6)^2$

Hint

PST in reverse: $(a + b)^2 = a^2 + 2ab + b^2$ .

Show answer

$x^2 + 12x + 36$ .

6. $(5x - 2)(-3x + 4)$

Hint

Box method; watch signs. Middle term is $+20x + 6x$ .

Show answer

$-15x^2 + 26x - 8$ .

	$-3x$	$+4$
$5x$	$-15x^2$	$+20x$
$-2$	$+6x$	$-8$

Sum: $-15x^2 + 20x + 6x - 8 = -15x^2 + 26x - 8$ .

Part B — Linear equations and inequalities

Solve for the variable. Watch the inequality-flip in #5, #6, #7, #8.

1. $3(x - 4) = 2x + 5$

Hint

Distribute first, then collect $x$ ‘s.

Show answer

$x = 17$ .

2. $5 - 2(y + 3) = 4y$

Hint

After distributing: $-1 - 2y = 4y$ , so $-1 = 6y$ .

Show answer

$y = -\tfrac{1}{6}$ .

3. $\dfrac{2x - 1}{3} = x - 4$

Hint

Multiply both sides by $3$ first to clear the fraction.

Show answer

$x = 11$ .

4. $4(2x + 1) - 3 = 5x + 10$

Hint

Distribute: $8x + 4 - 3 = 5x + 10 \implies 8x + 1 = 5x + 10$ .

Show answer

$x = 3$ .

5. $-3x + 7 \geq 19$

Hint

Isolate $-3x \geq 12$ , then divide by $-3$ . What happens to the inequality?

Show answer

$x \leq -4$ . Dividing both sides by $-3$ flips the inequality.

6. $-2(y - 5) < 8$

Hint

$-2y + 10 < 8 \implies -2y < -2$ . Divide by $-2$ (flip!).

Show answer

$y > 1$ .

7. $7 - 4x \leq 3x + 35$

Hint

Get $x$ ‘s on one side: $-7x \leq 28$ . Divide by $-7$ (flip).

Show answer

$x \geq -4$ .

8. $\dfrac{1 - 2y}{4} > 3$

Hint

Multiply both sides by $4$ first, then isolate $y$ . Don’t forget to flip when dividing by a negative.

Show answer

$y < -\tfrac{11}{2}$ .

Self-check question

When do you flip the inequality sign? (Whenever you multiply or divide both sides by a negative number — never when adding or subtracting.)

Thursday — Mixed Part-I Timed Set

Objective

Simulate Part I conditions and stress-test the heuristics under time pressure.

Instructions

Set a timer for 18 minutes.
Answer all 12 questions below. Show your scratch work to the right.
Don’t peek at hints or answers during the timed set. After the timer ends, check each problem one at a time. For every miss, write a Mistake Log entry tagged with the H-bucket that would have caught it.

Practice set

1. The expression $\dfrac{12}{\sqrt{3}}$ is equivalent to: (1) $\sqrt{3}$ (2) $4\sqrt{3}$ (3) $12\sqrt{3}$ (4) $36$

Hint

Multiply numerator and denominator by $\sqrt{3}$ to rationalize.

Show answer

(2) $4\sqrt{3}$ . $\tfrac{12}{\sqrt{3}} \cdot \tfrac{\sqrt{3}}{\sqrt{3}} = \tfrac{12\sqrt{3}}{3} = 4\sqrt{3}$ .

2. The graph of $f(x) = x^2 - 4$ has a minimum at: (1) $(0, -4)$ (2) $(-4, 0)$ (3) $(2, 0)$ (4) $(0, 4)$

Hint

$y = x^2$ has vertex at $(0, 0)$ ; subtracting 4 shifts it.

Show answer

(1) $(0, -4)$ .

3. Which scenario is exponential? (1) saves $10 each week (2) earns 8% interest each year (3) loses 5 lbs per month (4) drives 60 mph

Hint

Exponential = constant percent change per period (not a constant amount).

Show answer

(2) Earns 8% interest each year.

4. Which polynomial has degree 4 and leading coefficient $-3$ ? (1) $4x^3 - 3x$ (2) $-3x^4 + 2x^2$ (3) $3x^4 - x$ (4) $-4x^3 + 3$

Hint

Degree = highest exponent; leading coefficient is the number in front of that term.

Show answer

(2) $-3x^4 + 2x^2$ .

5. Simplify $(2x^2 + 3) - (x^2 - 5x + 1)$ . (1) $x^2 - 5x + 2$ (2) $x^2 + 5x + 2$ (3) $x^2 + 5x + 4$ (4) $3x^2 + 5x + 4$

Hint

Distribute the minus across all three terms of the second polynomial.

Show answer

(2) $x^2 + 5x + 2$ .

6. The function $g(x)$ is shown with values: $g(1) = 3,\ g(3) = 11,\ g(5) = 27$ . The average rate of change from $x = 1$ to $x = 5$ is: (1) 6 (2) 7 (3) 8 (4) 24

Hint

Average rate of change = $\dfrac{g(5) - g(1)}{5 - 1}$ .

Show answer

(1) 6. $\tfrac{27 - 3}{5 - 1} = \tfrac{24}{4} = 6$ .

7. Which is equivalent to $a^6 - b^4$ ? (1) $(a^3 - b^2)(a^3 + b^2)$ (2) $(a^2 - b^2)^3$ (3) $(a^3)^2 - b^4$ only (4) cannot be factored

Hint

$a^6 = (a^3)^2$ and $b^4 = (b^2)^2$ — that’s a difference of squares.

Show answer

(1) $(a^3 - b^2)(a^3 + b^2)$ .

8. Solve for $y$ : $4y - 7 = 2y + 11$ . (1) $y = 2$ (2) $y = 4$ (3) $y = 9$ (4) $y = -9$

Hint

Collect $y$ ‘s on one side: $2y = 18$ .

Show answer

(3) $y = 9$ .

9. Which represents $f(x) = |x|$ shifted 3 units left? (1) $|x| + 3$ (2) $|x| - 3$ (3) $|x + 3|$ (4) $|x - 3|$

Hint

Inside the function, " $+3$ " shifts the graph left. Counterintuitive but true.

Show answer

(3) $|x + 3|$ .

10. The domain of the graph that is open at $(-2, 1)$ and closed at $(6, 9)$ is: (1) $[1, 9]$ (2) $(1, 9]$ (3) $[-2, 6]$ (4) $(-2, 6]$

Hint

Domain is the $x$ -range. Open dot → strict; closed dot → inclusive.

Show answer

(4) $(-2, 6]$ .

11. Solve $\dfrac{3(x + 2)}{2} = x + 6$ . (1) $x = 6$ (2) $x = 4$ (3) $x = 2$ (4) $x = 0$

Hint

Multiply both sides by $2$ first: $3(x + 2) = 2x + 12$ .

Show answer

(1) $x = 6$ .

12. Which sequence is geometric with $a_1 = 8$ and common ratio $\tfrac{1}{2}$ ? (1) $8, 16, 32, 64, \ldots$ (2) $8, 4, 2, 1, \ldots$ (3) $8, 10, 12, 14, \ldots$ (4) $8, 0, -8, -16, \ldots$

Hint

Geometric means multiply by $r$ each step. $r = \tfrac{1}{2}$ means each term is half the previous.

Show answer

(2) $8, 4, 2, 1, \ldots$

Self-grade

11–12 correct: on target. 8–10: review the misses and the H-bucket. <8: redo Mon and Tue before continuing.

Friday — Re-do Q13 and Q30, plus Sibling Problems

Objective

Convert the actual June 25 misses into wins, with extra siblings to prove the fix is durable.

Instructions

For each problem, write the H2 rationale next to your work (“This is a domain question because…” / “This is a difference of squares because…”). Then solve, peek at the hint only if needed, and reveal the answer last.

Q13 (original)

A graph runs from open $(0, 15)$ to closed $(12, 45)$ . State the domain.

Hint

Domain is the $x$ -range. Open on the left → strict; closed on the right → inclusive.

Show answer

$0 < x \leq 12$ .

Wrong choice on June 25: $15 < x \leq 45$ — that’s the range (y-coords), not the domain.

Q13 siblings

13a. Graph open at $(-7, 1)$ , closed at $(3, 21)$ .

Hint

Domain uses $x$ -coords; open ↔ strict $<$ .

Show answer

Domain: $-7 < x \leq 3$ . Range: $1 < y \leq 21$ .

13b. Graph closed at $(-1, -1)$ , open at $(5, 11)$ .

Hint

Closed on left → $\leq$ ; open on right → strict $<$ .

Show answer

Domain: $-1 \leq x < 5$ . Range: $-1 \leq y < 11$ .

13c. $f(x) = \sqrt{x - 4}$ .

Hint

You need $x - 4 \geq 0$ for the square root to be real.

Show answer

Domain: $x \geq 4$ . Range: $y \geq 0$ .

13d. Horizontal segment $y = 8$ from $x = -3$ to $x = 7$ , both endpoints closed.

Hint

Horizontal segment → range is a single value.

Show answer

Domain: $-3 \leq x \leq 7$ . Range: $\{8\}$ .

Q30 (original)

Factor $x^3 - 36x$ completely.

Hint

Pull a GCF first; what’s left should be a DOS.

Show answer

$x(x - 6)(x + 6)$ .

Mistake on June 25: wrote $x(x - 6)^2$ . That’s a different pattern (PST), and multiplying it back out gives $x^3 - 12x^2 + 36x$ , not $x^3 - 36x$ .

Q30 siblings

30a. $x^3 - 49x$

Hint

Pull $x$ , then DOS on $x^2 - 49$ .

Show answer

$x(x - 7)(x + 7)$ .

30b. $4x^3 - 100x$

Hint

GCF is $4x$ , leaving $x^2 - 25$ .

Show answer

$4x(x - 5)(x + 5)$ .

30c. $2y^4 - 50y^2$

Hint

GCF is $2y^2$ , leaving $y^2 - 25$ .

Show answer

$2y^2(y - 5)(y + 5)$ .

30d. $x^5 - x$

Hint

GCF $x$ , then $x^4 - 1$ is DOS — and the result has another DOS inside.

Show answer

$x(x - 1)(x + 1)(x^2 + 1)$ .

Step-by-step: $x^5 - x = x(x^4 - 1) = x(x^2 - 1)(x^2 + 1) = x(x - 1)(x + 1)(x^2 + 1)$ . The $x^2 + 1$ doesn’t factor over the reals.

Self-check question

What two patterns are needed to factor $x^5 - x$ completely? (GCF, then DOS applied twice.)

Saturday — Full Part I + Part II Timed

Objective

Measure the week’s progress under real test conditions.

Instructions

Use a prior Regents Algebra I administration (any year; June 2024 or January 2025 are good). NYS releases them at https://www.nysedregents.org.
Print out Part I (Q1–Q24) and Part II (Q25–Q30) only.
Set timer for 70 minutes. Work in pen.
Score yourself strictly against the official answer key.
Compare scores to baseline (June 25 was Part I = 44/48, Part II = 10/12).

Success bar for Week 1

Part I score: 45+/48.
Part II score: 11+/12.
Every miss in the Mistake Log has an H-bucket tag.
Can name the structure of 9/10 factoring problems within 5 seconds.

Mistake Log review

At the end of the day, flip through every entry from the week. Group them by H-bucket. The H-bucket with the most entries is your top priority going into Week 2.

← Back to main study plan | → Week 2