How do you determine the number of intersection points between a line y = mx + d and a parabola y = ax^2 + bx + c?

Set the two equations equal (ax^2 + bx + c = mx + d), move all terms to one side to form a single quadratic ax^2 + (b - m)x + (c - d) = 0, and evaluate its Discriminant D = B^2 - 4AC: • D > 0 => 2 intersections • D = 0 => 1 intersection (tangent) • D 0 intersections.

How do the degree n and leading coefficient a determine the End Behavior of a polynomial P(x) = a*x^n + ...?

• Even degree, a > 0: Both ends up (as x -> ±∞, y -> +∞). • Even degree, a 0: Left end down, right end up. • Odd degree, a < 0: Left end up, right end down.

What are the six core Laws of Exponents and Radicals?

1. x^a * x^b = x^(a + b) 2. x^a / x^b = x^(a - b) 3. (x^a)^b = x^(a * b) 4. (xy)^a = x^a * y^a 5. x^(-a) = 1 / x^a 6. x^(m / n) = n-th root of (x^m) = (n-th root of x)^m.

What are the three core Logarithm Rules tested on the ACT Math section?

Definition: log_b(x) = y b^y = x. 1. Product Rule: log_b(MN) = log_b(M) + log_b(N) 2. Quotient Rule: log_b(M/N) = log_b(M) - log_b(N) 3. Power Rule: log_b(M^k) = k * log_b(M) Change of Base: log_b(a) = log(a) / log(b).

How do you find the Vertical and Horizontal Asymptotes of a rational function f(x) = P(x) / Q(x) on the ACT?

• Vertical Asymptote(s): Set the simplified denominator Q(x) = 0 (after canceling any common factors, which create holes). • Horizontal Asymptote: Compare degrees of P (num) and Q (den): if deg(P) deg(Q), no horizontal asymptote.

How does the multiplicity (exponent) of a root (x - r)^k affect how a polynomial graph behaves at the x-axis?

• Odd multiplicity (k = 1, 3, ...): The graph CROSSES through the x-axis at x = r. • Even multiplicity (k = 2, 4, ...): The graph TOUCHES (is tangent to) the x-axis at x = r and turns around without crossing.

In the Exponential Growth/Decay model f(t) = a * b^(t / k), what do a, b, and k represent?

• a = initial value when t = 0. • b = growth/decay factor per interval (b = 1 + r for growth; b = 1 - r for decay; b = 2 for doubling; b = 0.5 for half-life). • k = length of time required for ONE full growth/decay period!

How do you rewrite an annual exponential growth equation P(t) = 1000(1.12)^t (where t is in years) to show the monthly growth rate?

Since 1 year = 12 months, replace t with (12t) and take the 12th root of the base so the value at t = 1 stays identical: P(t) = 1000 * (1.12^(1/12))^(12t) ≈ 1000 * (1.00949)^(12t).

What does the Polynomial Remainder Theorem state?

When a polynomial P(x) is divided by a linear binomial (x - a), the remainder R is equal to P(a). Consequently, (x - a) is a factor of P(x) if and only if P(a) = 0 (the Factor Theorem).

When an SAT problem gives two identical-looking polynomials on both sides of an equation (e.g., (ax + 3)(2x - b) = 10x^2 + cx - 12 for all x), what is the fastest strategy?

Expand the factored side and match the coefficients of corresponding powers of x (x^2 term to x^2 term, x term to x term, constant to constant): 2ax^2 + (6 - ab)x - 3b = 10x^2 + cx - 12 => 2a = 10 (a=5), -3b = -12 (b=4), c = 6 - 20 = -14.

How do you find the Vertex (h, k) and Axis of Symmetry of a parabola in Standard Form y = ax^2 + bx + c?

• Axis of symmetry (and x-coordinate of vertex): h = -b / (2a). • y-coordinate of vertex (minimum if a > 0, maximum if a < 0): k = f(-b / (2a)) = c - b^2 / (4a). • If given the x-intercepts r1 and r2, the vertex x-coordinate is their midpoint: h = (r1 + r2) / 2.

When a polynomial P(x) is divided by D(x) yielding quotient Q(x) and remainder R, how is the identity written?

P(x) / D(x) = Q(x) + R / D(x), or equivalently P(x) = D(x) * Q(x) + R. To find constants in rational equivalence problems on the SAT, either plug in strategic values of x (like x = 0 or x = 1) or clear denominators.

How do you solve for the minimum or maximum value of a quadratic word problem (e.g., projectile height or revenue maximization)?

Either find the vertex x-coordinate h = -b/(2a) and evaluate f(h), OR graph the function in Desmos and click the peak/trough of the parabola: • The x-coordinate of the vertex is the input (time or price) that maximizes/minimizes y. • The y-coordinate k of the vertex is the actual maximum or minimum value!

How does the sign of the Discriminant D = b^2 - 4ac determine the number of real solutions (x-intercepts) of a quadratic?

• D > 0: Exactly TWO distinct real solutions (graph crosses the x-axis twice). • D = 0: Exactly ONE real solution (a double root; vertex is tangent to the x-axis). • D < 0: ZERO real solutions (two complex conjugate roots; parabola never touches the x-axis).

How do you solve an exponential equation where both sides can be written with a common base (e.g., 9^(x+1) = 27^x)?

Rewrite both bases as powers of the same prime base (here 3): (3^2)^(x+1) = (3^3)^x => 3^(2x + 2) = 3^(3x). Equate the exponents: 2x + 2 = 3x => x = 2.

What causes an Extraneous Solution in a Rational Equation (an equation with variables in denominators)?

Any value of x that makes ANY denominator in the original equation equal to 0 is outside the domain (undefined) and must be rejected as an extraneous solution.

What causes an Extraneous Solution when solving a radical equation like sqrt(2x + 3) = x?

Squaring both sides of an equation can introduce an extraneous root where the right side was negative (since the principal square root sqrt(...) is ALWAYS >= 0). Always plug candidate solutions back into the ORIGINAL unsquared equation!

What is the formula for the n-th term and sum of an Arithmetic vs. Geometric Sequence on the ACT?

• Arithmetic (constant difference d): a_n = a_1 + (n - 1)d; Sum S_n = (n / 2)(a_1 + a_n). • Geometric (constant ratio r): a_n = a_1 * r^(n - 1); Infinite sum (for |r| < 1): S = a_1 / (1 - r).

How do you Complete the Square to convert x^2 + bx + c = 0 into Vertex Form or Circle Form?

Take half of the linear coefficient b, square it ((b/2)^2), and add it to both sides: x^2 + bx + (b/2)^2 = (x + b/2)^2.

How do vertical stretches and compressions y = a * f(x) affect the zeros (x-intercepts) and y-intercept of a polynomial?

Multiplying f(x) by a non-zero constant 'a' scales the y-intercept and the vertex height by 'a', but does NOT change the x-intercepts (zeros) at all, because a * f(x) = 0 still has the exact same solutions as f(x) = 0.

How do you solve an SAT 'Equivalent Expressions' problem in 10 seconds using Desmos?

Graph the original expression as `f(x) = ...` in Desmos and graph the answer choices. The correct equivalent expression will trace the exact same curve on the coordinate plane for all values of x!

How do you evaluate a Composite Function (f ∘ g)(x) = f(g(x))?

Work from the INSIDE out: first evaluate the inner function g(x) at the given input, then take that output and plug it in as the input to the outer function f.

What are the three Special Factoring Identities tested constantly on the SAT and ACT?

1. Difference of Squares: a^2 - b^2 = (a - b)(a + b) 2. Square of a Binomial Sum: (a + b)^2 = a^2 + 2ab + b^2 3. Square of a Binomial Difference: (a - b)^2 = a^2 - 2ab + b^2.

What is the Quadratic Formula and what is the Discriminant D?

x = [-b ± sqrt(b^2 - 4ac)] / (2a). The Discriminant is the expression inside the radical: D = b^2 - 4ac.

If a bacterial colony doubles every 15 minutes and time t is measured in HOURS, vs. if t is measured in MINUTES, what is the exponent?

• If t is in minutes (and 1 period = 15 minutes): f(t) = a * 2^(t / 15). • If t is in hours (so 4 periods occur per 1 hour): f(t) = a * 2^(4t). Test plug-in: after 15 minutes (t = 0.25 hours), the exponent must equal 1!

If f(x) = a * b^x passes through (0, 5) and (2, 45) with b > 0, how do you find f(4)?

Since f(0) = 5, a = 5. At x = 2, 5 * b^2 = 45 => b^2 = 9 => b = 3. Every increase of 2 in x multiplies y by 9, so f(4) = 45 * 9 = 405.

How do you rationalize a denominator containing a square root or a binomial radical (e.g., 1 / (3 - sqrt(2)))?

• Single root 1/sqrt(a): multiply top and bottom by sqrt(a)/sqrt(a). • Binomial root 1/(a - sqrt(b)): multiply top and bottom by the conjugate (a + sqrt(b)) so the denominator becomes a difference of squares: a^2 - b.

How do you find the Inverse Function f^(-1)(x) of a one-to-one function y = f(x)?

Swap x and y in the equation, then solve algebraically for y. Geometrically, the graph of f^(-1)(x) is the reflection of f(x) across the line y = x, and if f(a) = b, then f^(-1)(b) = a.

What are Vieta's Formulas for the Sum and Product of the solutions to ax^2 + bx + c = 0?

• Sum of solutions (r1 + r2) = -b / a. • Product of solutions (r1 * r2) = c / a. High-yield SAT shortcut: Never use the full quadratic formula when a question asks for the sum or product of solutions—just calculate -b/a or c/a!

How do you solve a quadratic inequality like x^2 - 5x - 6 < 0?

Factor to find the boundary roots: (x - 6)(x + 1) 0), it is negative (< 0) BETWEEN its roots: -1 < x < 6.

What is the domain of f(x) = sqrt(g(x)) / h(x) in the real numbers?

Two simultaneous conditions must hold: 1. The radicand inside an even root must be non-negative: g(x) >= 0. 2. The denominator cannot equal zero: h(x) != 0.

What are the three algebraic forms of a Quadratic Function, and what geometric feature does each display as constants in the equation?

1. Standard Form: f(x) = ax^2 + bx + c -> displays the y-intercept (0, c). 2. Factored (Intercept) Form: f(x) = a(x - r1)(x - r2) -> displays the x-intercepts/roots (r1, 0) and (r2, 0). 3. Vertex Form: f(x) = a(x - h)^2 + k -> displays the vertex (minimum or maximum) at (h, k).

Passport to Advanced Math: Quadratics, Discriminant, Exponentials & Polynomials

Advanced SAT & ACT algebra: quadratic forms, vertex formulas, discriminant root analysis, Vieta's formulas, exponential growth/decay with fractional time intervals, rational equations, and polynomial remainder theorem.

Level: advanced
Languages: en → en
Cards: 32
Markdown: Passport to Advanced Math: Quadratics, Discriminant, Exponentials & Polynomials (.md)

Flashcards (32)

#	Front (Term / Question)	Back (Definition / Answer)
1	How do you determine the number of intersection points between a line y = mx + d and a parabola y = ax^2 + bx + c?	Set the two equations equal (ax^2 + bx + c = mx + d), move all terms to one side to form a single quadratic ax^2 + (b - m)x + (c - d) = 0, and evaluate its Discriminant D = B^2 - 4AC: • D > 0 => 2 intersections • D = 0 => 1 intersection (tangent) • D < 0 => 0 intersections.
2	How do the degree n and leading coefficient a determine the End Behavior of a polynomial P(x) = a*x^n + ...?	• Even degree, a > 0: Both ends up (as x -> ±∞, y -> +∞). • Even degree, a < 0: Both ends down. • Odd degree, a > 0: Left end down, right end up. • Odd degree, a < 0: Left end up, right end down.
3	What are the six core Laws of Exponents and Radicals?	1. x^a * x^b = x^(a + b) 2. x^a / x^b = x^(a - b) 3. (x^a)^b = x^(a * b) 4. (xy)^a = x^a * y^a 5. x^(-a) = 1 / x^a 6. x^(m / n) = n-th root of (x^m) = (n-th root of x)^m.
4	What are the three core Logarithm Rules tested on the ACT Math section?	Definition: log_b(x) = y <=> b^y = x. 1. Product Rule: log_b(MN) = log_b(M) + log_b(N) 2. Quotient Rule: log_b(M/N) = log_b(M) - log_b(N) 3. Power Rule: log_b(M^k) = k * log_b(M) Change of Base: log_b(a) = log(a) / log(b).
5	How do you find the Vertical and Horizontal Asymptotes of a rational function f(x) = P(x) / Q(x) on the ACT?	• Vertical Asymptote(s): Set the simplified denominator Q(x) = 0 (after canceling any common factors, which create holes). • Horizontal Asymptote: Compare degrees of P (num) and Q (den): if deg(P) < deg(Q), y = 0; if deg(P) = deg(Q), y = ratio of leading coefficients; if deg(P) > deg(Q), no horizontal asymptote.
6	How does the multiplicity (exponent) of a root (x - r)^k affect how a polynomial graph behaves at the x-axis?	• Odd multiplicity (k = 1, 3, ...): The graph CROSSES through the x-axis at x = r. • Even multiplicity (k = 2, 4, ...): The graph TOUCHES (is tangent to) the x-axis at x = r and turns around without crossing.
7	In the Exponential Growth/Decay model f(t) = a * b^(t / k), what do a, b, and k represent?	• a = initial value when t = 0. • b = growth/decay factor per interval (b = 1 + r for growth; b = 1 - r for decay; b = 2 for doubling; b = 0.5 for half-life). • k = length of time required for ONE full growth/decay period!
8	How do you rewrite an annual exponential growth equation P(t) = 1000(1.12)^t (where t is in years) to show the monthly growth rate?	Since 1 year = 12 months, replace t with (12t) and take the 12th root of the base so the value at t = 1 stays identical: P(t) = 1000 * (1.12^(1/12))^(12t) ≈ 1000 * (1.00949)^(12t).
9	What does the Polynomial Remainder Theorem state?	When a polynomial P(x) is divided by a linear binomial (x - a), the remainder R is equal to P(a). Consequently, (x - a) is a factor of P(x) if and only if P(a) = 0 (the Factor Theorem).
10	When an SAT problem gives two identical-looking polynomials on both sides of an equation (e.g., (ax + 3)(2x - b) = 10x^2 + cx - 12 for all x), what is the fastest strategy?	Expand the factored side and match the coefficients of corresponding powers of x (x^2 term to x^2 term, x term to x term, constant to constant): 2ax^2 + (6 - ab)x - 3b = 10x^2 + cx - 12 => 2a = 10 (a=5), -3b = -12 (b=4), c = 6 - 20 = -14.
11	How do you find the Vertex (h, k) and Axis of Symmetry of a parabola in Standard Form y = ax^2 + bx + c?	• Axis of symmetry (and x-coordinate of vertex): h = -b / (2a). • y-coordinate of vertex (minimum if a > 0, maximum if a < 0): k = f(-b / (2a)) = c - b^2 / (4a). • If given the x-intercepts r1 and r2, the vertex x-coordinate is their midpoint: h = (r1 + r2) / 2.
12	When a polynomial P(x) is divided by D(x) yielding quotient Q(x) and remainder R, how is the identity written?	P(x) / D(x) = Q(x) + R / D(x), or equivalently P(x) = D(x) * Q(x) + R. To find constants in rational equivalence problems on the SAT, either plug in strategic values of x (like x = 0 or x = 1) or clear denominators.
13	How do you solve for the minimum or maximum value of a quadratic word problem (e.g., projectile height or revenue maximization)?	Either find the vertex x-coordinate h = -b/(2a) and evaluate f(h), OR graph the function in Desmos and click the peak/trough of the parabola: • The x-coordinate of the vertex is the input (time or price) that maximizes/minimizes y. • The y-coordinate k of the vertex is the actual maximum or minimum value!
14	How does the sign of the Discriminant D = b^2 - 4ac determine the number of real solutions (x-intercepts) of a quadratic?	• D > 0: Exactly TWO distinct real solutions (graph crosses the x-axis twice). • D = 0: Exactly ONE real solution (a double root; vertex is tangent to the x-axis). • D < 0: ZERO real solutions (two complex conjugate roots; parabola never touches the x-axis).
15	How do you solve an exponential equation where both sides can be written with a common base (e.g., 9^(x+1) = 27^x)?	Rewrite both bases as powers of the same prime base (here 3): (3^2)^(x+1) = (3^3)^x => 3^(2x + 2) = 3^(3x). Equate the exponents: 2x + 2 = 3x => x = 2.
16	What causes an Extraneous Solution in a Rational Equation (an equation with variables in denominators)?	Any value of x that makes ANY denominator in the original equation equal to 0 is outside the domain (undefined) and must be rejected as an extraneous solution.
17	What causes an Extraneous Solution when solving a radical equation like sqrt(2x + 3) = x?	Squaring both sides of an equation can introduce an extraneous root where the right side was negative (since the principal square root sqrt(...) is ALWAYS >= 0). Always plug candidate solutions back into the ORIGINAL unsquared equation!
18	What is the formula for the n-th term and sum of an Arithmetic vs. Geometric Sequence on the ACT?	• Arithmetic (constant difference d): a_n = a_1 + (n - 1)d; Sum S_n = (n / 2)(a_1 + a_n). • Geometric (constant ratio r): a_n = a_1 * r^(n - 1); Infinite sum (for \|r\| < 1): S = a_1 / (1 - r).
19	How do you Complete the Square to convert x^2 + bx + c = 0 into Vertex Form or Circle Form?	Take half of the linear coefficient b, square it ((b/2)^2), and add it to both sides: x^2 + bx + (b/2)^2 = (x + b/2)^2.
20	How do vertical stretches and compressions y = a * f(x) affect the zeros (x-intercepts) and y-intercept of a polynomial?	Multiplying f(x) by a non-zero constant 'a' scales the y-intercept and the vertex height by 'a', but does NOT change the x-intercepts (zeros) at all, because a * f(x) = 0 still has the exact same solutions as f(x) = 0.
21	How do you solve an SAT 'Equivalent Expressions' problem in 10 seconds using Desmos?	Graph the original expression as `f(x) = ...` in Desmos and graph the answer choices. The correct equivalent expression will trace the exact same curve on the coordinate plane for all values of x!
22	How do you evaluate a Composite Function (f ∘ g)(x) = f(g(x))?	Work from the INSIDE out: first evaluate the inner function g(x) at the given input, then take that output and plug it in as the input to the outer function f.
23	What are the three Special Factoring Identities tested constantly on the SAT and ACT?	1. Difference of Squares: a^2 - b^2 = (a - b)(a + b) 2. Square of a Binomial Sum: (a + b)^2 = a^2 + 2ab + b^2 3. Square of a Binomial Difference: (a - b)^2 = a^2 - 2ab + b^2.
24	What is the Quadratic Formula and what is the Discriminant D?	x = [-b ± sqrt(b^2 - 4ac)] / (2a). The Discriminant is the expression inside the radical: D = b^2 - 4ac.
25	If a bacterial colony doubles every 15 minutes and time t is measured in HOURS, vs. if t is measured in MINUTES, what is the exponent?	• If t is in minutes (and 1 period = 15 minutes): f(t) = a * 2^(t / 15). • If t is in hours (so 4 periods occur per 1 hour): f(t) = a * 2^(4t). Test plug-in: after 15 minutes (t = 0.25 hours), the exponent must equal 1!
26	If f(x) = a * b^x passes through (0, 5) and (2, 45) with b > 0, how do you find f(4)?	Since f(0) = 5, a = 5. At x = 2, 5 * b^2 = 45 => b^2 = 9 => b = 3. Every increase of 2 in x multiplies y by 9, so f(4) = 45 * 9 = 405.
27	How do you rationalize a denominator containing a square root or a binomial radical (e.g., 1 / (3 - sqrt(2)))?	• Single root 1/sqrt(a): multiply top and bottom by sqrt(a)/sqrt(a). • Binomial root 1/(a - sqrt(b)): multiply top and bottom by the conjugate (a + sqrt(b)) so the denominator becomes a difference of squares: a^2 - b.
28	How do you find the Inverse Function f^(-1)(x) of a one-to-one function y = f(x)?	Swap x and y in the equation, then solve algebraically for y. Geometrically, the graph of f^(-1)(x) is the reflection of f(x) across the line y = x, and if f(a) = b, then f^(-1)(b) = a.
29	What are Vieta's Formulas for the Sum and Product of the solutions to ax^2 + bx + c = 0?	• Sum of solutions (r1 + r2) = -b / a. • Product of solutions (r1 * r2) = c / a. High-yield SAT shortcut: Never use the full quadratic formula when a question asks for the sum or product of solutions—just calculate -b/a or c/a!
30	How do you solve a quadratic inequality like x^2 - 5x - 6 < 0?	Factor to find the boundary roots: (x - 6)(x + 1) < 0, so roots are x = -1 and x = 6. Since the parabola opens upward (a > 0), it is negative (< 0) BETWEEN its roots: -1 < x < 6.
31	What is the domain of f(x) = sqrt(g(x)) / h(x) in the real numbers?	Two simultaneous conditions must hold: 1. The radicand inside an even root must be non-negative: g(x) >= 0. 2. The denominator cannot equal zero: h(x) != 0.
32	What are the three algebraic forms of a Quadratic Function, and what geometric feature does each display as constants in the equation?	1. Standard Form: f(x) = ax^2 + bx + c -> displays the y-intercept (0, c). 2. Factored (Intercept) Form: f(x) = a(x - r1)(x - r2) -> displays the x-intercepts/roots (r1, 0) and (r2, 0). 3. Vertex Form: f(x) = a(x - h)^2 + k -> displays the vertex (minimum or maximum) at (h, k).