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Advanced Math

Quadratics · Polynomials · Rational · Radical
CORD — C Concept O Operation · Desmos R Recognition D Drill
R1 Quadratic — vertex, intercepts, min/max Drill 11
Desmos
  1. 3 points → Quadratic Regression: table + y_1 ~ a x_1^2 + b x_1 + c → reads a, b, c.
  2. Equation given: type any form — y = x^2 - 6x + 5 or y = (x - 1)(x - 5).
  3. Click the peak / valley dot → vertex $(p, q)$. min/max value $= q$.
  4. Click x-axis dots → roots. y-axis dot → y-intercept.
Lesson · Quadratic Regression →
Concept
  • Vieta's: for $ax^2 + bx + c = 0$, sum $= -b/a$, product $= c/a$.
  • Vertex form $a(x-p)^2 + q$ → vertex $(p, q)$ directly.
  • Build from vertex + 1 point: plug in vertex, solve $a$.
SAT answers may be fractions or radicals. $1.414 = \sqrt{2}$ — convert.
RecognitionChooses a rounded decimal when choices require exact radical form.
R2 Quadratic — tangent line & # of solutions Drill 56
Desmos
  1. Plot both curves: y = x^2 and y = kx + 1.
  2. Desmos prompts to add a slider for the unknown.
  3. Drag until the line just touches the parabola → read value.
  4. "How many solutions" → count intersection dots.
  5. Quadratic through 3 points: table → Quadratic Regression → reads $a, b, c$.
Lesson · Quadratic Regression →
Concept
  • Set equal: $ax^2 + bx + c = mx + n$ → $ax^2 + (b-m)x + (c-n) = 0$.
  • Discriminant $D = b^2 - 4ac$: $D < 0$ no sol · $D = 0$ tangent (1 sol) · $D > 0$ two sols.
RecognitionThinks D = 0 gives two roots; it gives one repeated root.
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Advanced Math

Polynomial & Rational
R3 Polynomial · degree $\geq 3$ Drill 14
Desmos
  1. Type p(x) = x^3 + 3x^2 - x - 3.
  2. Click each x-intercept → roots.
  3. Remainder when $p(x) \div (x - a)$: type p(a) → reads value.
  4. $(x-a)$ is a factor $\Leftrightarrow p(a) = 0$.
Lesson · Cubic Factoring →
Concept
  • Remainder theorem: $p(x) \div (x-a)$ leaves $p(a)$. If $p(a) = 0$, then $(x-a)$ is a factor.
  • Odd multiplicity → crosses. Even → bounces.
  • Vieta's for cubic $ax^3 + bx^2 + cx + d$: sum of roots $= -b/a$, product $= -d/a$.
RecognitionTreats p(a) as a factor instead of the remainder.
R4 Rational & square-root Drill 14
Desmos
  1. Type the function: y = (x^2 - 4) / (x - 2) or y = sqrt(x - 3) + 1.
  2. Vertical asymptote = the gap in the curve.
  3. Horizontal asymptote = the $y$ the curve flattens toward.
  4. Hole = small gap where numerator + denominator share a factor.
Lesson · Rational & Square-Root →
Concept
  • $\dfrac{a}{x - h} + k$: vertical $x = h$, horizontal $y = k$.
  • $\sqrt{x - a} + b$: domain $x \geq a$, starts $(a, b)$.
  • Holes: factor, cancel; the cancelled value is the hole.
RecognitionMisses the hole by cancelling without noting the excluded x-value.
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Advanced Math

Exponential — y = ab^x · Modeling
R5 Exponential — identify & build from data Drill 10
Desmos
  1. Enter points as a table (+ icon → table).
  2. Pick Exponential Regression: y_1 ~ ab^{x_1}.
  3. Desmos prints $a$ and $b$ directly.
  4. To verify a candidate: graph it, check it hits the given points.
Lesson · Exponential Regression →
Concept
  • Table: constant ratio → exponential. Constant 2nd diff → quadratic.
  • $y = ab^x$: y-intercept $= a$.
  • $b > 1$ growth, $0 < b < 1$ decay.
RecognitionReads y-intercept as b instead of a in y=ab^x.
R6 Exponential modeling — growth, decay, equivalent forms Drill 30
Desmos
  1. Type the model: f(t) = 50000 * 1.04^t.
  2. Value at $t = N$: type f(N) on a new line.
  3. "When does $y$ reach $X$": add y = X, click intersection.
  4. "Equivalent expression": graph both → check curves overlap.
Lesson · Exponential Regression →
Concept
  • Doubling every $T$: $y = a \cdot 2^{t/T}$.
  • Half-life $H$: $y = a \cdot (1/2)^{t/H}$.
  • Grows $r\%$/year: $y = a(1 + r)^t$.
  • Decays $r\%$/year: $y = a(1 - r)^t$.
  • Substitution trick: for problems like $f(x) = 4^x + a \cdot 2^{x+2} + 3$:
    let $2^x = t$ → becomes quadratic in $t$. Solve in $t$, then convert back.
"Increases 4%" means $b = 1.04$, not $0.04$. The "$1+$" is the #1 trap.
RecognitionReads '4% increase' as b=0.04 instead of b=1.04.
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Advanced Math

Function — Evaluation · Transformations · Word setup
R7 Function evaluation & transformations Drill 28
Desmos
  1. Define f(x) = ..., then type f(5) → reads value.
  2. Composition: type f(g(2)).
  3. Plot original + transformed → see the shift visually.
Lesson · Function Transformations →
Concept
  • $f(x) + c$ → shifts UP by $c$.
  • $f(x + c)$ → shifts LEFT by $c$ (positive $c$ → left).
  • $-f(x)$: reflects over x-axis.   $f(-x)$: reflects over y-axis.
  • $a \cdot f(x)$: vertical stretch ($|a| > 1$) or shrink ($< 1$).
  • Composition $f(g(x))$: plug $g(x)$ wherever $x$ appears.
$f(x + 3)$ shifts LEFT 3. Inside the parens = opposite direction.
RecognitionReads f(x+3) as shifting right 3 instead of left 3.
R8 Word setup & Rearrange formula Drill 34
Desmos
  1. After setup, plot the equation.
  2. "Object hits ground" → click positive x-intercept.
  3. "Max revenue / height" → click peak.
Concept
  • Area: $L \cdot W = A$. "length 5 more than width" → $x(x + 5) = A$.
  • Projectile: $h(t) = -16 t^2 + v_0 t + h_0$ (ft/sec). Meters: $-4.9$.
  • Revenue: price $\times$ quantity. Maximize at vertex.
  • Rearrange formula: "Express $q$ in terms of $r, s$" → isolate $q$ using inverse ops (add/sub, mul/div, square/$\sqrt{\,}$).
Rearrange: divide by a variable only if you know it's nonzero. Square-root step: include both $\pm$ branches.
RecognitionDrops the negative branch after square-rooting when rearranging a formula.
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Advanced Math

Algebra mechanics — Simplify · Factor
R9 Simplify polynomial — combine + distribute + FOIL Drill 34
Desmos
  1. Type LHS into slot 1, candidate answer into slot 2.
  2. If graphs overlap exactly → they're equivalent.
  3. Or evaluate both at $x = 2, x = -1, x = 3$ — same value at all 3 → equivalent (degree $\leq 2$).
Concept
  • Combine like: same variable + same power only. $3x^2 + 5x^2 = 8x^2$.
  • Square binomial: $(a+b)^2 = a^2 + 2ab + b^2$. NOT $a^2 + b^2$.
  • Diff of squares: $(a+b)(a-b) = a^2 - b^2$.
  • Difference of squares from sum: $(a+b)^2 - (a-b)^2 = 4ab$.
  • Sum of squares trick: $a^2 + b^2 = (a+b)^2 - 2ab$.
  • Telescoping product: $(a-b)(a+b)(a^2+b^2)(a^4+b^4) = a^8 - b^8$.
    e.g. $(a+2)(a-2)(a^2+4)(a^4+16)$ collapses to $a^8 - 256$.
Most common error: $(x+3)^2 = x^2 + 9$ ❌. Forgot the $2 \cdot x \cdot 3 = 6x$ cross-term.
RecognitionExpands (x+3)² as x²+9 and forgets the 6x cross-term.
R10 Factor polynomial — GCF / diff-of-squares / trinomial Drill 19
Desmos
  1. Plot the polynomial. Click each x-intercept to read roots $r_1, r_2, \dots$.
  2. Factored form: $a(x - r_1)(x - r_2)\dots$ where $a$ is the leading coefficient.
  3. For $x^2 + bx + c$: roots from graph give factors directly.
  4. Overlap check: graph original AND each answer-choice factored form — the one whose curve exactly overlaps is correct.
Lesson · Cubic Factoring →
Always click the grey dot — never eyeball. College Board plants distractor roots 0.5 off the visual estimate. Click for exact value.
Concept
  • Step 1 — GCF first: $2x^3 + 6x^2 = 2x^2(x + 3)$.
  • Diff of squares: $a^2 - b^2 = (a-b)(a+b)$. e.g. $9x^2 - 25 = (3x-5)(3x+5)$.
  • Trinomial $x^2 + bx + c$: find two numbers that multiply to $c$ and add to $b$.
  • Trinomial $ax^2 + bx + c$: use quadratic formula OR AC-grouping.
  • Perfect square trinomial: $a^2 + 2ab + b^2 = (a+b)^2$.
Always pull GCF first — otherwise you'll miss a factor or get wrong roots.
RecognitionMisses a difference-of-squares like x²-16 and tries to factor it as a trinomial.
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Advanced Math

Exponents · Rational expressions · Solving
R11 Exponent rules & Rational expressions Drill 20
Desmos
  1. If all variables are defined numerically: type the expression — Desmos shows the value (expand row if needed).
  2. For symbolic equivalence: pick test values like $x = 2, -1, 3$, evaluate both forms — if same value at all 3, equivalent.
  3. Rational $\frac{f(x)}{g(x)}$: graph to spot common factors as removable holes.
Concept
  • Multiply same base: $x^a \cdot x^b = x^{a+b}$.
  • Divide same base: $x^a / x^b = x^{a-b}$.
  • Power of power: $(x^a)^b = x^{ab}$.
  • Negative exponent: $x^{-a} = 1/x^a$.
  • Zero exponent: $x^0 = 1$.
  • Fractional exponent: $x^{a/b} = \sqrt[b]{x^a} = (\sqrt[b]{x})^a$.
  • Add rational: common denom. $\tfrac{1}{a} + \tfrac{1}{b} = \tfrac{a+b}{ab}$.
  • Simplify rational: factor numerator + denominator, cancel common.
$(x+y)^2 \neq x^2 + y^2$  and  $\sqrt{x^2 + y^2} \neq x + y$. No distributing exponents over sums.
RecognitionDistributes the exponent over a sum, treating (x+y)² as x²+y².
R12 Solve nonlinear equation — quadratic / abs / radical / rational Drill 44
Desmos
  1. Vector regression for non-linear: for $f(x) = g(x)$, type [f(x_1)] ~ [g(x_1)]. Desmos prints $x_1$ as regression parameter. Then $y = g(x_1)$. Example: $\sqrt{2x+4} = 2x+2$ → [sqrt(2x_1+4)] ~ [2x_1+2] → $x_1 = 0$, $y = 2$.
  2. Works for radical, rational, quadratic — exact value even if intersection is off-screen.
  3. Backup (standard): move all to one side $f(x) = $ LHS $-$ RHS, plot, click x-intercept. OR plot both, click intersection.
  4. Absolute value: type |x - a| = b as-is → vertical lines pop up at solutions.
Lesson · Vector Regression →
Lesson · Non-Linear System (click) →
Lesson · Absolute Value →
Concept
  • Quadratic formula: $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
  • Zero product: $(x-a)(x-b) = 0 \to x = a$ or $x = b$.
  • Radical: isolate $\sqrt{\dots}$, square both sides, then check for extraneous roots.
  • Rational: multiply both sides by LCD, then solve. Reject any solution making a denominator $= 0$.
  • Absolute: $|x - a| = b \to x - a = b$ or $x - a = -b$.
Squaring & multiplying by variable can introduce extraneous roots — always plug back to verify.
RecognitionForgets to check for extraneous roots after squaring or clearing denominators.
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