The function \( f(x) = 500 \cdot (0.8)^x \) models exponential ___.
Identify the base
Base = 0.8 → Is it greater than 1 or less than 1?
Less than 1 → the value gets smaller each step → growth or decay?
A Growth, because 500 is large
B Decay, because the base 0.8 is less than 1
C Growth, because the exponent x is positive
D Neither — it is linear
Q 05Algebra IITrickyRational Expressions
🔑KEY: excluded values — set the denominator = 0 and solve; those x-values are EXCLUDED
For the expression \(\dfrac{x+3}{x^2 - 9}\), which values of \( x \) are excluded?
Factor the denominator first
\( x^2 - 9 = (x-3)(x+3) \)
Set each factor = 0: \( x = 3 \) and \( x = -3 \)
Both are excluded — even though \( x+3 \) also cancels!
A Only \( x = 3 \)
B Only \( x = -3 \)
C \( x = 3 \) and \( x = -3 \)
D No excluded values
Q 06Algebra IIMediumInverse Functions
🔑KEY: swap x and y, then solve for y — that's the inverse!
Find the inverse of \( f(x) = 3x - 6 \).
Steps
Step 1: Write \( y = 3x - 6 \)
Step 2: Swap x and y → \( x = 3y - 6 \)
Step 3: Solve for y → \( y = \dfrac{x+6}{3} \)
A \( f^{-1}(x) = \dfrac{x-6}{3} \)
B \( f^{-1}(x) = \dfrac{x+6}{3} \)
C \( f^{-1}(x) = 3x + 6 \)
D \( f^{-1}(x) = \dfrac{1}{3x-6} \)
Q 07Algebra IITrickySequences — Arithmetic vs Geometric
Test: divide consecutive terms
Geometric → each term ÷ previous term = SAME ratio
Arithmetic → each term − previous term = SAME difference
A \( 3,\ 7,\ 11,\ 15,\ \ldots \)
B \( 2,\ 5,\ 8,\ 11,\ \ldots \)
C \( 4,\ 12,\ 36,\ 108,\ \ldots \)
D \( 1,\ 4,\ 9,\ 16,\ \ldots \)
Q 08Algebra IIMediumPolynomial — End Behavior
🔑KEY: leading term decides end behavior — check degree (even/odd) and leading coefficient (+ / −)
Describe the end behavior of \( f(x) = -3x^4 + 2x - 1 \) as \( x \to +\infty \).
Two-step check
Leading term: \( -3x^4 \)
Even degree + negative leading coeff → both ends go DOWN
As \( x \to +\infty \), \( f(x) \to \) ?
A \( f(x) \to +\infty \)
B \( f(x) \to -\infty \)
C \( f(x) \to 0 \)
D \( f(x) \to +\infty \) on left, \( -\infty \) on right
Q 09Algebra IITrickyRadical Equations
🔑KEY: after solving, always CHECK for extraneous solutions by plugging back in!
Solve: \(\sqrt{2x + 3} = x - 1\). Which value(s) of \( x \) are valid solutions?
Solve & Check
Square both sides: \( 2x+3 = (x-1)^2 = x^2-2x+1 \)
→ \( x^2 - 4x - 2 = 0 \)... wait, let's try: rearrange → \(x^2 -4x -2=0\). Actually check \(x=6\) and \(x=-1\): plug back into original to see which works!
A \( x = -1 \) only
B \( x = 6 \) only
C \( x = 6 \) and \( x = -1 \)
D No solution
Q 10Algebra IIMediumVertex Form of Parabola
🔑KEY: vertex form: y = a(x − h)² + k → vertex is (h, k). Watch the SIGN of h!
What is the vertex of \( f(x) = 2(x + 4)^2 - 7 \)?
Trap: sign of h!
Form is \( a(x - h)^2 + k \), so \( (x+4) = (x-(-4)) \)
→ \( h = -4 \), \( k = -7 \) → vertex = ?
Common mistake: writing (+4, −7) instead of (−4, −7)
A \( (4,\ -7) \)
B \( (4,\ 7) \)
C \( (-4,\ -7) \)
D \( (-4,\ 7) \)
Advanced Geometry — 10 Questions
Q 11GeometryMediumCircle — Arc Length
🔑KEY: Arc Length = (central angle / 360) × 2πr — it's a FRACTION of the full circumference
A circle has radius \( 9 \). What is the arc length for a central angle of \( 80° \)? (Leave answer in terms of \( \pi \))
A sphere has a radius of \( 5 \) cm. What is its surface area? (Use \( \pi \approx 3.14 \))
Formula
SA = \( 4\pi r^2 = 4(3.14)(5^2) = 4(3.14)(25) = ? \) Trap: Students often use \( 2\pi r^2 \) (that's a hemisphere!) or confuse with volume formula.
🔑KEY: CPCTC = Corresponding Parts of Congruent Triangles are Congruent — used AFTER proving triangles ≅
In a proof, you've shown that \( \triangle ABD \cong \triangle CBD \). To conclude that \( \angle A \cong \angle C \), you use ___.
Order matters in proofs
Step 1: Prove the triangles are congruent (SAS, ASA, SSS, AAS…)
Step 2: THEN use CPCTC to say any matching parts are equal
You CANNOT use CPCTC as a reason to prove congruence — only after!