Passport to Advanced Math

Ultra exam-oriented. Master proportional reasoning and data inference. Expect multi-step setups and interpretation under time pressure.

Passport to Advanced Math

Exam-oriented practice on nonlinear functions and expressions: quadratics, exponentials, radicals, rational expressions, and function structure.

Advanced Math Mastery: Functions & Quadratics

Concept Forms & Algebraic Cues 800-Level Strategic Insights
Quadratic Structures
\(y = a(x-h)^2 + k\)

\(x_v = -\frac{b}{2a}\)
 VERTEX FORM
The SAT often asks for the maximum or minimum value (this is always \(k\)). If the problem asks for the \(x\)-value where it occurs, that is \(h\). Tip: If roots are \(r_1\) and \(r_2\), the vertex \(h\) is the midpoint: \(\frac{r_1+r_2}{2}\).
Root Theorems
\(\text{Sum} = -\frac{b}{a}\)

\(\text{Product} = \frac{c}{a}\)
 VIETA’S FORMULAS
Use these to solve for constants without solving the full quadratic. Discriminant (\(\Delta\)): If a line and parabola intersect once, set equations equal and force \(b^2 – 4ac = 0\).
Exponential Modeling
\(y = a(b)^{\frac{t}{k}}\)
 GROWTH/DECAY
\(a\) is initial value; \(b\) is growth factor. If it “doubles every 3 hours,” \(b=2\) and \(k=3\). Key: If the rate is given as a percentage increase \(r\), then \(b = 1 + r\). For decrease, \(b = 1 – r\).
Rational Functions
\(f(x) = \frac{P(x)}{Q(x)}\)
 ASYMPTOTES
Vertical: Where \(Q(x) = 0\) (after canceling holes).
Horizontal: If degrees are equal, \(y = \frac{\text{lead coeff}}{\text{lead coeff}}\). If degree of numerator is smaller, \(y = 0\).
Function Notation
\(f(g(x))\)

\(f(x-h) + k\)
 TRANSFORMATIONS
Inside: Horizontal shifts (opposite direction). \(f(x-3)\) moves 3 units right.
Outside: Vertical shifts. \(f(x)+5\) moves 5 units up.
Inverse: To find \(f^{-1}(x)\), reflect the graph over the line \(y = x\).

Advanced Math: Mastery Worked Examples (15)

Quadratics & Parabolas

1. Hard

A parabola in the xy-plane is given by the equation \( y = a(x-3)^2 + k \). If the parabola passes through the points \((1, -4)\) and \((6, 11)\), what is the value of \( a + k \)?

Show Detailed Solution

1. **Set up equations:** Substitute both points into the vertex form.
– For \((1, -4)\): \( -4 = a(1-3)^2 + k \Rightarrow -4 = 4a + k \)
– For \((6, 11)\): \( 11 = a(6-3)^2 + k \Rightarrow 11 = 9a + k \)
2. **Solve the system:** Subtract the first equation from the second.
– \( (11 – (-4)) = (9a – 4a) + (k – k) \Rightarrow 15 = 5a \Rightarrow a = 3 \).
3. **Find \(k\):** Plug \(a=3\) back into the first eq: \( -4 = 4(3) + k \Rightarrow -4 = 12 + k \Rightarrow k = -16 \).
4. **Final Step:** \( a + k = 3 + (-16) = -13 \).

Answer: -13

2. Ivy

The function \( f(x) = x^2 + bx + c \) has roots \( r_1 \) and \( r_2 \). If \( r_1 + r_2 = 10 \) and \( r_1 \cdot r_2 = 16 \), what is the minimum value of the function \( f(x) \)?

Show Detailed Solution

1. **Determine the Equation:** Using Vieta’s formulas, \( b = -(r_1+r_2) = -10 \) and \( c = r_1 \cdot r_2 = 16 \). Thus, \( f(x) = x^2 – 10x + 16 \).
2. **Find the Vertex x-coordinate:** \( x_v = -b/(2a) = 10/2 = 5 \).
3. **Find the Minimum Value (y-coordinate):** Plug \( x=5 \) into the function.
– \( f(5) = (5)^2 – 10(5) + 16 = 25 – 50 + 16 = -9 \).

Answer: -9

3. MIT

For what value of \( c \) does the system \( \begin{cases} y = x^2 – 4x + 7 \\ y = 2x + c \end{cases} \) have exactly one solution?

Show Detailed Solution

1. **Equate the functions:** \( x^2 – 4x + 7 = 2x + c \).
2. **Rearrange into standard form:** \( x^2 – 6x + (7 – c) = 0 \).
3. **Apply Tangency Condition:** For one solution, the discriminant (\(b^2 – 4ac\)) must be 0.
– \( a = 1, b = -6, c’ = (7 – c) \).
– \( (-6)^2 – 4(1)(7 – c) = 0 \Rightarrow 36 – 28 + 4c = 0 \).
4. **Solve:** \( 8 + 4c = 0 \Rightarrow 4c = -8 \Rightarrow c = -2 \).

Answer: -2

Exponential & Radical Functions

4. Hard

Solve for \( x \): \( \sqrt{2x + 15} – x = 6 \).

Show Detailed Solution

1. **Isolate the radical:** \( \sqrt{2x + 15} = x + 6 \).
2. **Square both sides:** \( 2x + 15 = (x + 6)^2 \Rightarrow 2x + 15 = x^2 + 12x + 36 \).
3. **Set to zero:** \( x^2 + 10x + 21 = 0 \).
4. **Factor:** \( (x + 3)(x + 7) = 0 \). Potential solutions: \( x = -3, x = -7 \).
5. **Check Extraneous Roots:**
– Test \( x = -3 \): \( \sqrt{2(-3)+15} – (-3) = \sqrt{9} + 3 = 6 \). (Valid)
– Test \( x = -7 \): \( \sqrt{2(-7)+15} – (-7) = \sqrt{1} + 7 = 8 \neq 6 \). (Invalid)

Answer: -3

5. Ivy

An isotope decays such that every 10 years, the mass is reduced by 15%. If the initial mass is 200g, what is the mass after 40 years?

Show Detailed Solution

1. **Determine the factor:** A 15% reduction means 85% remains. Multiplier = \( 0.85 \).
2. **Find number of cycles:** \( 40 \text{ years} / 10 \text{ years per cycle} = 4 \text{ cycles} \).
3. **Calculate mass:** \( \text{Mass} = 200(0.85)^4 \).
4. **Compute:** \( (0.85)^2 = 0.7225 \); \( (0.7225)^2 \approx 0.522 \).
5. **Final:** \( 200 \times 0.522 = 104.4 \text{g} \).

Answer: 104.4g

Rational Functions & Notation

6. MIT

If \( f(x) = \dfrac{x+3}{x-2} \), find the expression for the inverse function \( f^{-1}(x) \).

Show Detailed Solution

1. **Replace notation:** Let \( y = \frac{x+3}{x-2} \).
2. **Switch x and y:** \( x = \frac{y+3}{y-2} \).
3. **Multiply to clear denominator:** \( x(y-2) = y+3 \Rightarrow xy – 2x = y + 3 \).
4. **Isolate y-terms:** \( xy – y = 2x + 3 \).
5. **Factor out y:** \( y(x – 1) = 2x + 3 \).
6. **Solve:** \( y = \frac{2x+3}{x-1} \).

Answer: \( f^{-1}(x) = \frac{2x+3}{x-1} \)

7. Hard

Find the vertical and horizontal asymptotes of \( g(x) = \dfrac{3x^2 – 12}{x^2 – x – 6} \).

Show Detailed Solution

1. **Simplify first:** Factor numerator and denominator.
– \( g(x) = \frac{3(x-2)(x+2)}{(x-3)(x+2)} \).
2. **Cancel common factors:** \( (x+2) \) cancels. There is a **hole** at \( x=-2 \).
3. **Vertical Asymptote:** Set remaining denominator to zero: \( x – 3 = 0 \Rightarrow x = 3 \).
4. **Horizontal Asymptote:** Degrees are equal (2 and 2). Ratio of leading coefficients: \( y = 3/1 = 3 \).

Vertical: \( x=3 \); Horizontal: \( y=3 \)

8. Ivy

If \( f(x-2) = x^2 + 5 \), what is the value of \( f(3) \)?

Show Detailed Solution

1. **Find the input value:** We want the “inside” of \( f(\dots) \) to be 3.
2. **Set up equation:** \( x – 2 = 3 \Rightarrow x = 5 \).
3. **Plug this x into the expression:** \( f(3) = (5)^2 + 5 = 25 + 5 = 30 \).

Answer: 30

9. MIT

Find the remainder when \( p(x) = 2x^3 – 5x^2 + x – 7 \) is divided by \( x – 3 \).

Show Detailed Solution

1. **Apply Remainder Theorem:** The remainder is simply \( p(3) \).
2. **Calculate:** \( p(3) = 2(3)^3 – 5(3)^2 + (3) – 7 \).
– \( 2(27) – 5(9) + 3 – 7 = 54 – 45 + 3 – 7 = 5 \).

Answer: 5

Advanced Algebraic Structures

10. Hard

If \( (ax+2)(bx+7) = 15x^2 + cx + 14 \) for all values of \(x\), and \( a+b = 8 \), what are the two possible values for \(c\)?

Show Detailed Solution

1. **Expand LHS:** \( abx^2 + 7ax + 2bx + 14 = abx^2 + (7a+2b)x + 14 \).
2. **Match coefficients:** \( ab = 15 \) and \( 7a+2b = c \).
3. **Solve for a and b:** We need two numbers that multiply to 15 and add to 8 (\(a+b=8\)).
– Case 1: \( a=3, b=5 \). Then \( c = 7(3) + 2(5) = 21 + 10 = 31 \).
– Case 2: \( a=5, b=3 \). Then \( c = 7(5) + 2(3) = 35 + 6 = 41 \).

Answer: 31 and 41

11. Ivy

Simplify: \( \dfrac{1}{x+2} + \dfrac{3}{x-5} \).

Show Detailed Solution

1. **Find LCD:** \( (x+2)(x-5) \).
2. **Rewrite numerators:** \( \frac{1(x-5) + 3(x+2)}{(x+2)(x-5)} \).
3. **Simplify numerator:** \( x – 5 + 3x + 6 = 4x + 1 \).
4. **Final Result:** \( \frac{4x+1}{x^2 – 3x – 10} \).

Answer: \( \frac{4x+1}{x^2 – 3x – 10} \)

12. MIT

If \( \sqrt{-1} = i \), what is the value of \( (3 + 2i)(1 – 4i) \)?

Show Detailed Solution

1. **FOIL:** \( 3(1) + 3(-4i) + 2i(1) + 2i(-4i) \).
2. **Expand:** \( 3 – 12i + 2i – 8i^2 \).
3. **Substitute \(i^2 = -1\):** \( 3 – 10i – 8(-1) = 3 – 10i + 8 = 11 – 10i \).

Answer: \( 11 – 10i \)

13. Hard

The graph of \( y = x^2 + 6x + 5 \) has a vertex at \((h, k)\). What is the value of \( k \)?

Show Detailed Solution

1. **Find h:** \( h = -b/(2a) = -6/2 = -3 \).
2. **Find k:** Plug \(x = -3\) into equation.
– \( k = (-3)^2 + 6(-3) + 5 = 9 – 18 + 5 = -4 \).

Answer: -4

14. Ivy

If \( 3^{x+2} = 81^x \), what is the value of \(x\)?

Show Detailed Solution

1. **Common Base:** Write 81 as \( 3^4 \).
2. **Equation:** \( 3^{x+2} = (3^4)^x \Rightarrow 3^{x+2} = 3^{4x} \).
3. **Equate Exponents:** \( x + 2 = 4x \).
4. **Solve:** \( 3x = 2 \Rightarrow x = 2/3 \).

Answer: 2/3

15. MIT

Find the value of \( a \) such that \( x+2 \) is a factor of \( x^3 + ax^2 – 4x + 12 \).

Show Detailed Solution

1. **Factor Theorem:** If \( x+2 \) is a factor, then \( p(-2) = 0 \).
2. **Substitute:** \( (-2)^3 + a(-2)^2 – 4(-2) + 12 = 0 \).
3. **Calculate:** \( -8 + 4a + 8 + 12 = 0 \Rightarrow 4a + 12 = 0 \).
4. **Solve:** \( 4a = -12 \Rightarrow a = -3 \).

Answer: -3

Elite Practice: Advanced Math Mastery (30 Questions)

Quadratics & Polynomials (Hard)
  1. 1. Hard If \( f(x) = x^2 – 12x + k \) has exactly one real root, what is the value of \( k \)?
    Show Detailed Solution

    1. Concept: One real root means the discriminant (\(b^2 – 4ac\)) must be zero.
    2. Identify: \( a=1, b=-12, c=k \).
    3. Solve: \( (-12)^2 – 4(1)(k) = 0 \Rightarrow 144 – 4k = 0 \Rightarrow k = 36 \).
    4. Verification: \( (x-6)^2 = x^2 – 12x + 36 \), which has one root at \( x=6 \).

  2. 2. Hard The sum of the roots of the equation \( 3x^2 + bx – 12 = 0 \) is 4. What is the value of \( b \)?
    Show Detailed Solution

    1. Vieta’s Formula: Sum of roots \( = -b/a \).
    2. Substitute: \( -b/3 = 4 \).
    3. Solve: \( -b = 12 \Rightarrow b = -12 \).

  3. 3. Hard Find the vertex of the parabola defined by \( y = -2(x+3)(x-5) \).
    Show Detailed Solution

    1. Find x-intercepts: \( x = -3 \) and \( x = 5 \).
    2. Find h (midpoint): \( (-3 + 5) / 2 = 1 \).
    3. Find k: Plug \( x=1 \) into the equation: \( y = -2(1+3)(1-5) = -2(4)(-4) = 32 \).
    4. Result: Vertex is \( (1, 32) \).

  4. 4. Hard If \( (x-k) \) is a factor of \( x^2 – 5x – 14 \), and \( k > 0 \), find \( k \).
    Show Detailed Solution

    1. Factor the quadratic: \( (x-7)(x+2) \).
    2. Compare: The factors are \( (x-7) \) and \( (x+2) \).
    3. Select k: Since \( k > 0 \), \( k = 7 \).

  5. 5. Hard What is the product of the solutions to \( 2x^2 + 10x – 48 = 0 \)?
    Show Detailed Solution

    1. Vieta’s Formula: Product of roots \( = c/a \).
    2. Identify: \( a=2, c=-48 \).
    3. Calculate: \( -48/2 = -24 \).

Ivy League Level (Conceptual Logic)
  1. 6. Ivy If \( f(x-3) = 2x^2 + x \), what is the value of \( f(2) \)?
    Show Detailed Solution

    1. Set inside equal: \( x – 3 = 2 \).
    2. Solve for x: \( x = 5 \).
    3. Plug into expression: \( 2(5)^2 + 5 = 2(25) + 5 = 55 \).

  2. 7. Ivy A parabola passes through \( (2, 0) \) and \( (8, 0) \). What is the x-coordinate of its vertex?
    Show Detailed Solution

    1. Symmetry: The vertex of a parabola always lies exactly halfway between its x-intercepts.
    2. Calculate: \( (2 + 8) / 2 = 5 \).

  3. 8. Ivy If \( f(x) = \frac{x+1}{x-2} \), what value of \( x \) is excluded from the domain of \( f^{-1}(x) \)?
    Show Detailed Solution

    1. Note: The excluded value in the inverse’s domain is the **Horizontal Asymptote** of the original function.
    2. Find H.A.: Ratio of leading coefficients is \( 1/1 = 1 \).
    3. Result: \( x=1 \).

  4. 9. Ivy A circle has the equation \( x^2 + y^2 + 10x – 4y = 7 \). What is the radius?
    Show Detailed Solution

    1. Complete the Square: \( (x+5)^2 + (y-2)^2 = 7 + 25 + 4 \).
    2. Simplify RHS: \( (x+5)^2 + (y-2)^2 = 36 \).
    3. Radius: \( \sqrt{36} = 6 \).

  5. 10. Ivy For the polynomial \( p(x) \), \( p(4) = 0 \). Which of the following must be a factor of \( p(x) \)?
    Show Detailed Solution

    1. Factor Theorem: If \( p(c) = 0 \), then \( (x-c) \) is a factor.
    2. Result: \( (x-4) \).

  6. 11. Ivy Simplify \( \frac{x^2 – 16}{x^2 – 2x – 8} \).
    Show Detailed Solution

    1. Factor numerator: \( (x-4)(x+4) \).
    2. Factor denominator: \( (x-4)(x+2) \).
    3. Cancel common: \( \frac{x+4}{x+2} \).

  7. 12. Ivy If \( 2^{x+3} = 32 \), what is the value of \( 3^{x-1} \)?
    Show Detailed Solution

    1. Solve for x: \( 2^{x+3} = 2^5 \Rightarrow x+3 = 5 \Rightarrow x = 2 \).
    2. Plug in: \( 3^{2-1} = 3^1 = 3 \).

  8. 13. Ivy Solve for \( x \): \( \sqrt{2x+6} + 4 = 10 \).
    Show Detailed Solution

    1. Isolate radical: \( \sqrt{2x+6} = 6 \).
    2. Square: \( 2x + 6 = 36 \).
    3. Solve: \( 2x = 30 \Rightarrow x = 15 \).

  9. 14. Ivy If \( i^2 = -1 \), what is the value of \( (2+3i)(2-3i) \)?
    Show Detailed Solution

    1. Difference of Squares: \( 2^2 – (3i)^2 \).
    2. Calculate: \( 4 – 9i^2 = 4 – 9(-1) = 13 \).

  10. 15. Ivy Find the horizontal asymptote of \( y = \frac{5x^3 – 2}{2x^3 + 7x} \).
    Show Detailed Solution

    1. Rule: If degrees match, ratio of leading coefficients.
    2. Calculate: \( y = 5/2 = 2.5 \).

MIT Level (Competitive Reasoning)
  1. 16. MIT If \( \alpha \) and \( \beta \) are roots of \( x^2 – 5x + 2 = 0 \), find \( \frac{1}{\alpha} + \frac{1}{\beta} \).
    Show Detailed Solution

    1. Combine: \( \frac{\alpha + \beta}{\alpha\beta} \).
    2. Use Vieta: \( \alpha + \beta = 5 \), \( \alpha\beta = 2 \).
    3. Result: \( 5/2 = 2.5 \).

  2. 17. MIT Find \( k \) so that \( y = 2x + k \) is tangent to \( y = x^2 \).
    Show Detailed Solution

    1. Set equal: \( x^2 – 2x – k = 0 \).
    2. Discriminant = 0: \( (-2)^2 – 4(1)(-k) = 0 \Rightarrow 4 + 4k = 0 \Rightarrow k = -1 \).

  3. 18. MIT Solve for \( x \): \( 9^x – 10(3^x) + 9 = 0 \).
    Show Detailed Solution

    1. Substitute: Let \( u = 3^x \). Equation becomes \( u^2 – 10u + 9 = 0 \).
    2. Factor: \( (u-9)(u-1) = 0 \Rightarrow u=9, u=1 \).
    3. Solve for x: \( 3^x = 9 \Rightarrow x=2 \); \( 3^x = 1 \Rightarrow x=0 \).

  4. 19. MIT If \( f(g(x)) = x \) and \( g(x) = \frac{x-1}{2} \), find \( f(x) \).
    Show Detailed Solution

    1. Note: \( f \) is the inverse of \( g \).
    2. Find inverse: \( x = \frac{y-1}{2} \Rightarrow 2x = y-1 \Rightarrow y = 2x+1 \).
    3. Result: \( f(x) = 2x+1 \).

  5. 20. MIT Find the remainder when \( x^{2024} + 1 \) is divided by \( x + 1 \).
    Show Detailed Solution

    1. Remainder Theorem: Plug in \( x = -1 \).
    2. Calculate: \( (-1)^{2024} + 1 = 1 + 1 = 2 \).

  6. 21. MIT If \( x + \frac{1}{x} = 4 \), find \( x^2 + \frac{1}{x^2} \).
    Show Detailed Solution

    1. Square both sides: \( (x + \frac{1}{x})^2 = 16 \).
    2. Expand: \( x^2 + 2(x)(\frac{1}{x}) + \frac{1}{x^2} = 16 \).
    3. Simplify: \( x^2 + 2 + \frac{1}{x^2} = 16 \Rightarrow x^2 + \frac{1}{x^2} = 14 \).

  7. 22. MIT Solve for \( x \): \( \log_2(x) + \log_2(x-2) = 3 \).
    Show Detailed Solution

    1. Combine logs: \( \log_2(x^2 – 2x) = 3 \).
    2. Exponentiate: \( x^2 – 2x = 2^3 = 8 \).
    3. Solve: \( x^2 – 2x – 8 = 0 \Rightarrow (x-4)(x+2) = 0 \).
    4. Domain check: \( x \) must be \( > 2 \), so \( x = 4 \).

  8. 23. MIT A parabola \( y = ax^2 + bx + c \) has vertex \( (0,0) \) and passes through \( (2,8) \). Find \( a \).
    Show Detailed Solution

    1. Vertex form: \( y = a(x-0)^2 + 0 \Rightarrow y = ax^2 \).
    2. Plug in point: \( 8 = a(2^2) \Rightarrow 8 = 4a \Rightarrow a = 2 \).

  9. 24. MIT If \( f(x) = \sqrt{x^2 – 9} \), what is the domain of \( f \)?
    Show Detailed Solution

    1. Inequality: \( x^2 – 9 \ge 0 \Rightarrow x^2 \ge 9 \).
    2. Solve: \( |x| \ge 3 \Rightarrow x \ge 3 \) or \( x \le -3 \).

  10. 25. MIT Find the sum of the infinite series \( 1 + 1/2 + 1/4 + 1/8 + \dots \).
    Show Detailed Solution

    1. Formula: \( S = a / (1-r) \).
    2. Identify: \( a = 1, r = 1/2 \).
    3. Calculate: \( 1 / (1 – 0.5) = 1 / 0.5 = 2 \).

  11. 26. MIT If \( x^2 + y^2 = 25 \) and \( xy = 12 \), find \( (x+y)^2 \).
    Show Detailed Solution

    1. Identity: \( (x+y)^2 = x^2 + y^2 + 2xy \).
    2. Substitute: \( 25 + 2(12) = 25 + 24 = 49 \).

  12. 27. MIT Solve \( \frac{1}{x} + \frac{1}{2x} = 3 \).
    Show Detailed Solution

    1. Common denom: \( \frac{2 + 1}{2x} = 3 \Rightarrow \frac{3}{2x} = 3 \).
    2. Solve: \( 3 = 6x \Rightarrow x = 0.5 \).

  13. 28. MIT What is the slope of the line tangent to \( f(x) = x^2 \) at \( x = 3 \)?
    Show Detailed Solution

    1. Power Rule (Calculus shortcut): \( f'(x) = 2x \).
    2. Plug in: \( 2(3) = 6 \). (SAT note: You can also use the discriminant of a system to find this without calculus).

  14. 29. MIT If \( f(x) = 3^{x} \), what is \( f(x+2) / f(x) \)?
    Show Detailed Solution

    1. Expression: \( 3^{x+2} / 3^x \).
    2. Exponent Rule: \( 3^{(x+2) – x} = 3^2 = 9 \).

  15. 30. MIT Solve for \( x \): \( |2x – 5| = |x + 4| \).
    Show Detailed Solution

    1. Case 1: \( 2x – 5 = x + 4 \Rightarrow x = 9 \).
    2. Case 2: \( 2x – 5 = -(x + 4) \Rightarrow 2x – 5 = -x – 4 \Rightarrow 3x = 1 \Rightarrow x = 1/3 \).

Next Steps

Repeat methods under time. Target: medium (<40s); advanced (<90s); accuracy ≥ 80%. Then continue to Additional Topics in Math.

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