PreCalc 20 — Master Quiz

01

Functions Hard

⚡ Quick Memory Point

f(x−h) → shift RIGHT h | f(x+h) → shift LEFT h
−f(x) → reflect OVER x-axis | f(−x) → reflect OVER y-axis

A water park designs a slide whose height (in meters) above the ground is modeled by h(t) = −2(t − 3)² + 18, where t is seconds after launch. A second slide is built with the model g(t) = h(−t + 5).

What single transformation maps h to g, and what is g's maximum height?

Hint: rewrite g(t) by substituting (−t + 5) into h.

📘 Worked Mini-Example

If f(x)=x², then f(−x+2) = (−x+2)² = (x−2)² — reflected over y-axis AND shifted right 2. Always factor out the negative first!

02

Domain & Range Hard

⚡ Quick Memory Point

EVEN root → radicand ≥ 0 | Denominator → ≠ 0
Combine restrictions with AND (intersection)

A biologist models population density as f(x) = √(4x − 8)x² − 9 where x represents kilometers from a river.

What is the domain of f(x)?

03

Inverse Functions Hard

⚡ Quick Memory Point

f⁻¹(f(x)) = x | Swap x↔y then solve for y
Domain of f = Range of f⁻¹ (and vice versa)

A chemist converts Celsius to Kelvin with K(C) = C + 273.15. She then applies a pressure-scaling model: P(K) = 3K² − 5 for K ≥ 0.

What is (P ∘ K)⁻¹(x)? (the inverse of the composite function)

04

Polynomial Roots Hard

⚡ Quick Memory Point

Multiplicity EVEN → TOUCH x-axis (bounce)
Multiplicity ODD → CROSS x-axis (pass through)

An engineer models beam deflection with D(x) = −2x(x − 4)²(x + 1)³ where x is position in meters.

At which x-values does the beam cross the x-axis (deflection changes sign), and at which does it only touch?

05

Remainder Theorem Medium-Hard

⚡ Quick Memory Point

Remainder Theorem: f(a) = remainder when f(x) ÷ (x−a)
Factor Theorem: f(a)=0 ↔ (x−a) is a factor

A factory's profit (in thousands) over t years is modeled by P(t) = 2t³ − 7t² + kt − 3. When the factory was 2 years old, the profit remainder upon dividing by (t − 2) was 5.

Find k, then determine whether (t − 2) is a factor of P(t).

06

Polynomial Inequalities Hard

⚡ Quick Memory Point

Sign chart: find zeros → test intervals → check ≥ or >
Always include endpoints when ≥ or ≤

A projectile's altitude (m) satisfies A(t) = −t³ + 6t² − 9t.

During which time interval(s) is A(t) ≥ 0? (i.e., the projectile is at or above ground)

07

Asymptotes Hard

⚡ Quick Memory Point

HA: compare degrees (top vs bottom)
deg↑>deg↓ → oblique asy (do long division!)
VA: denominator = 0 AND doesn't cancel

A drone's speed (m/s) relative to wind resistance is modeled by S(w) = 2w² − 3w + 1w − 2.

Find all asymptotes of S(w) and classify each.

08

Rational Inequalities Hard

⚡ Quick Memory Point

NEVER multiply both sides by variable (sign unknown!)
Move everything to ONE side → sign chart

A car's fuel efficiency (km/L) must satisfy: x + 3x − 1 > 2

Solve the inequality for x.

09

Exponential Growth Hard

⚡ Quick Memory Point

Doubling time T: A = A₀·2^(t/T)
Continuous: A = A₀·eᵏᵗ | k = ln(2)/T

A bacterial colony starts with 500 cells and doubles every 3 hours. A second colony starts 6 hours later with 4000 cells and grows continuously at a rate of k = ln(2)/3 per hour.

After how many hours from the start (t = 0) do both colonies have equal population?

10

Logarithm Laws Hard

⚡ Quick Memory Point

log(AB)=logA+logB | log(A/B)=logA−logB
log(Aⁿ)=n·logA | Change of base: log_b(x)=ln(x)/ln(b)

An earthquake's Richter magnitude is given by M = log₁₀(I/I₀). Earthquake A is magnitude 5.2 and earthquake B is magnitude 7.8.

How many times MORE intense is earthquake B compared to earthquake A? (exact form)

10^(7.8 − 5.2) = 10^2.6 ≈ ?

11

Log Equations Hard

⚡ Quick Memory Point

ALWAYS check: argument of log must be > 0
Extraneous solutions often appear — verify!

A sound engineer uses: log₂(x − 3) + log₂(x + 1) = 5 to model audio signal x (watts).

Solve for x and state whether any solutions must be rejected.

12

Sinusoidal Models Hard

⚡ Quick Memory Point

y = A·sin(Bx−C)+D | A=amplitude | Period=2π/B
Phase shift = C/B | Midline = D

Ocean tides at a harbor follow: max depth = 12 m at 2:00 AM, min depth = 4 m at 8:00 AM.

Write the sinusoidal model D(t), then find the depth at t = 5 AM.

13

Trig Identities Hard

⚡ Quick Memory Point

sin²θ+cos²θ=1 → sin²θ=1−cos²θ
Work on ONE side only. Convert to sin/cos first.

A physics student needs to simplify a wave equation and encounters: 1 − cos²θsinθ + sin²θ1 + cosθ

What does this expression simplify to?

14

Trig Equations Hard

⚡ Quick Memory Point

General solution: sin=k → θ = arcsin(k) + 2πn OR π−arcsin(k) + 2πn
cos=k → θ = ±arccos(k) + 2πn

An electrical engineer solves for current zero-crossings using: 2sin²x − sinx − 1 = 0, where 0 ≤ x < 2π.

Find all solutions in [0, 2π).

15

Geometric Series Hard

⚡ Quick Memory Point

Infinite Geo Sum: S = a/(1−r), only if |r| < 1
Finite: Sₙ = a(1−rⁿ)/(1−r)

A bouncing ball drops from 10 m. Each bounce reaches 60% of the previous height.

What is the TOTAL distance the ball travels (up AND down), including the initial drop?

16

Arithmetic Series Medium-Hard

⚡ Quick Memory Point

Sₙ = n/2·(first + last) = n/2·(2a+(n−1)d)
aₙ = a + (n−1)d

A construction company lays bricks in rows. Row 1 has 3 bricks, row 2 has 7, row 3 has 11, and so on (arithmetic).

If the total number of bricks used is 820, how many rows are there?

17

Ellipse Hard

⚡ Quick Memory Point

Ellipse: x²/a² + y²/b² = 1 | c² = a² − b² (a > b)
Foci at (±c, 0) if horizontal | SUM of distances to foci = 2a

A satellite orbit is modeled as an ellipse with the equation (x−2)²25 + (y+1)²9 = 1 (in thousands of km).

Find the coordinates of the foci of this orbit.

18

Hyperbola Hard

⚡ Quick Memory Point

Hyperbola: x²/a²−y²/b²=1 → opens LEFT/RIGHT
Asymptotes: y = ±(b/a)x | c²=a²+b²

Two radio towers are 26 km apart and located at the foci of a hyperbola. A ship receives signals such that the difference in distances to the two towers is always 10 km.

Write the standard form equation of the hyperbola (center at origin, foci on x-axis).

19

Binomial Theorem Hard

⚡ Quick Memory Point

General term: T(r+1) = C(n,r)·aⁿ⁻ʳ·bʳ
To find specific term: set the power you want → solve for r

A pharmaceutical company models drug concentration using the expansion of (2x − 1x)⁸.

Find the term that is independent of x (the constant term) in the expansion.

T(r+1) = C(8,r) · (2x)^(8−r) · (−1/x)^r

20

Permutations & Combinations Hard

⚡ Quick Memory Point

ORDER matters → Permutation P(n,r)=n!/(n−r)!
ORDER doesn't matter → Combination C(n,r)=n!/r!(n−r)!

A committee of 5 must be chosen from 7 men and 6 women. The committee must include at least 2 women.

How many such committees are possible?

💡 Strategy Hint

Use complementary counting OR add up cases: exactly 2W, exactly 3W, exactly 4W, exactly 5W. Cases: (2W+3M), (3W+2M), (4W+1M), (5W+0M). Add all.