Troy School District · Mathematics · Algebra 2 Honors
Form A · 120 minutes · 100 points

Algebra 2 Honors — Testing-Out Examination

A student who passes this examination has demonstrated mastery of the Common Core Algebra 2 Honors standards (CCSS-M conceptual categories N-CN, A-APR, A-REI, F-IF, F-BF, F-LE, F-TF, S-IC) at the rigor of a final exam in a standard honors textbook (Larson, Algebra 2 Honors; Forester, Algebra and Trigonometry) and is eligible to advance directly to AP Precalculus.

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Instructions

Part I · No Calculator · Problems 1–5 · 35 pts · ~45 min
1. Polynomial structure: factoring, sign analysis, end behavior 8 points

Let \( P(x) = x^4 - 2x^3 - 11x^2 + 12x + 36 \).

  1. Use the Rational Root Theorem to list all candidate rational zeros of \(P(x)\). Then verify, using synthetic division, that \(x = -2\) is a zero. Show the synthetic-division tableau.
  2. Using your work in (a), factor \(P(x)\) completely over the integers. (After dividing by \((x+2)\) you obtain a cubic; the cubic also has a rational root.)
  3. State the multiplicity of each real zero. Determine the end behavior of \(y = P(x)\) (i.e., the limits as \(x \to \pm\infty\)) and justify from the leading coefficient and degree.
  4. Sketch \(y = P(x)\) on a labeled coordinate plane. Mark every \(x\)-intercept with its multiplicity (a graph crosses the axis at odd multiplicity, touches and turns at even multiplicity).
  5. Solve the polynomial inequality \(P(x) \le 0\). Express the solution in interval notation; justify by sign analysis on the factored form.
2. Complex numbers and the Fundamental Theorem of Algebra 7 points
  1. Compute exactly. Write each in standard form \(a + bi\):

    (i) \( (3 - 2i)(1 + 4i) \)    (ii) \( \dfrac{4 + i}{2 - 3i} \) (rationalize the denominator)    (iii) \( i^{2027} \)

  2. A polynomial \(Q(x)\) of degree 4 with real coefficients and leading coefficient 1 has \(1 + 2i\) and \(\sqrt{3}\) among its zeros.
    1. List all four zeros of \(Q(x)\). Cite the conjugate-pair principle for real coefficients and the analogous principle for irrational conjugate pairs (when the polynomial is over \(\mathbb{Q}\)).
    2. Write \(Q(x)\) as a product of two real quadratic factors. Multiply each conjugate pair carefully and simplify.
  3. Suppose \(R(x)\) is a degree-3 polynomial with real coefficients and zeros \(1+2i\) and \(4\). Is \(R(x)\) determined uniquely up to leading coefficient? Justify in one sentence.
3. Composition, inverse, and domain reasoning 7 points

Let \( \displaystyle f(x) = \frac{2x - 3}{x + 1} \) and \( g(x) = \sqrt{\,x - 2\,} \).

  1. Compute \((f \circ g)(6)\) exactly. Show each substitution.
  2. Determine the domain of \(f \circ g\). Identify each excluded value and explain in one sentence per exclusion why it is excluded.
  3. Find \(f^{-1}(x)\) algebraically by solving \(y = f(x)\) for \(x\). State the domain and range of \(f^{-1}\) using the domain–range duality (do not solve a separate inequality).
  4. Verify directly using your formula in (c) that \(f\!\bigl(f^{-1}(5)\bigr) = 5\). Show every step.
4. Logarithm properties & equation solving (no calculator) 7 points
  1. Express \( \displaystyle \log_3\!\left(\frac{81}{\sqrt{3}\,\cdot\,9^{n}}\right) \) as a single number plus a multiple of \(n\). Cite each property of logarithms you use (product, quotient, power) by name.
  2. Solve for \(x\): \( \log_2 x + \log_2(x - 6) = 4 \). Show each algebraic step. State which solution(s) are valid; explain why any extraneous solution must be discarded.
  3. On the same axes, sketch \( y = \log_2 x \) and \( y = \log_2(x - 4) - 1 \). Label each vertical asymptote, each \(x\)-intercept, and one additional point on each curve. Then describe in one sentence the geometric transformation that takes the first graph to the second.
5. Trigonometric values and identity verification (no calculator) 6 points
  1. Find the exact value of \( \cos\!\left(\dfrac{7\pi}{12}\right) \). Identify the sum or difference identity you use, and explain in one sentence why your decomposition of \(\dfrac{7\pi}{12}\) is convenient (e.g., as \(\dfrac{\pi}{3} + \dfrac{\pi}{4}\)).
  2. Suppose \( \sin\theta = \dfrac{3}{5} \) and \( \theta \in \!\left(\dfrac{\pi}{2},\; \pi\right) \). Find the exact values of \( \cos\theta \), \( \sin(2\theta) \), and \( \cos(2\theta) \). Justify the sign of \(\cos\theta\) from the quadrant of \(\theta\).
  3. Verify the identity \( \displaystyle \frac{1 - \cos^2 x}{\sin x \,\cos x} = \tan x \) for all \(x\) in its common domain. Then state the values of \(x\) (in \([0, 2\pi)\)) for which either side is undefined, and confirm those exclusions are the same on both sides.
Part II · Graphing Calculator Permitted (no CAS) · Problems 6–12 · 65 pts · ~75 min
6. Rational function modeling — drug concentration 10 points

A simple one-compartment pharmacokinetic model gives the concentration \(C\) (in mg/L) of a drug in the bloodstream as a function of time \(t\) (in hours) since a single intravenous injection, \[ C(t) = \frac{12\,t}{t^2 + 4}, \qquad t \ge 0. \]

  1. Compute \(C(0)\), \(C(2)\), and \(C(10)\). Round each to two decimal places.
  2. Identify any horizontal asymptote of \(y = C(t)\). State the value and explain in one sentence what it tells you about the drug's long-term blood concentration.
  3. Show algebraically (without calculus) that \(C\) attains its maximum at \(t = 2\). Hint: rewrite \(C(t) - C(2)\) over a common denominator and factor; the sign of the resulting expression for \(t \ne 2\) settles the matter.
  4. The drug is therapeutically effective at concentrations of at least 2 mg/L. For approximately how long does the drug remain effective? Solve \(C(t) \ge 2\) algebraically; report the time interval to the nearest tenth of an hour. Note: a calculator-only "intersect" answer earns at most half credit; you must write and reduce the equation.
  5. Suppose a second injection of equal dose is administered at \(t = 5\), and the two drug processes superpose linearly. On a labeled axis, sketch — without computing exact values — the new total concentration \(C_{\text{tot}}(t)\) for \(0 \le t \le 14\). Justify the qualitative features (initial rise, two local maxima or one shoulder, eventual decay to 0).
7. Continuous exponential decay — caffeine 8 points

A 200 mg dose of caffeine is consumed at 8:00 a.m. The half-life of caffeine in an average adult is 5 hours. Let \(Q(t)\) denote the milligrams of caffeine remaining in the body \(t\) hours after consumption.

  1. Write a continuous-decay model \(Q(t) = Q_0\,e^{kt}\). Determine \(Q_0\) and \(k\) exactly; express \(k\) using \(\ln 2\).
  2. Using your model, how much caffeine remains at 6:00 p.m.? Round to the nearest tenth of a milligram.
  3. Caffeine is generally believed to impair sleep at body burdens above 50 mg. At what time of day does the burden first drop below 50 mg? Show the algebra; round to the nearest minute.
  4. The same person consumes a second 100 mg dose at 2:00 p.m. Write a piecewise function \(Q_{\text{tot}}(t)\) for the total caffeine in their body for \(0 \le t \le 18\), where \(t\) is hours after 8:00 a.m. Use your exact \(k\) from (a).
8. Logarithmic scales — earthquake magnitude 8 points

The moment-magnitude \(M\) of an earthquake is related to the energy \(E\) (in joules) it releases by \[ M = \tfrac{2}{3} \,\log_{10}\!\left(\frac{E}{E_0}\right), \qquad E_0 = 10^{4.4}\ \text{J}. \]

  1. Solve the relation for \(E\) in terms of \(M\). Simplify so that \(E\) is written as a single power of 10.
  2. The 1906 San Francisco earthquake measured \(M_{\text{SF}} = 7.9\). The 2011 Tōhoku earthquake measured \(M_{\text{T}} = 9.1\). Compute the ratio \(E_{\text{T}}/E_{\text{SF}}\). Show the algebra symbolically before evaluating; round the final number to the nearest integer.
  3. Explain in one or two complete sentences what your answer to (b) means physically.
  4. A scientist proposes redefining magnitude using natural logarithm: \( M' = \tfrac{2}{3}\, \ln(E/E_0) \). Express \(M'\) as a constant multiple of \(M\); that is, find \(c\) such that \(M' = cM\). Round \(c\) to four decimal places, and write one sentence on why \(c \ne 1\).
9. Sinusoidal modeling — Ferris wheel 10 points

A Ferris wheel has diameter 80 ft. Its lowest passenger position is 5 ft above the ground. The wheel rotates at a constant rate, completing one full revolution every 90 seconds. A passenger boards at the lowest point at \(t = 0\). Let \(h(t)\) be the passenger's height above the ground (in feet) at time \(t\) seconds after boarding.

  1. Identify the amplitude, period, midline, and starting position. Write \(h(t)\) explicitly as a cosine function of \(t\). Justify the sign you place on the cosine using the starting position.
  2. At what times during the first revolution (\(0 \le t \le 90\)) is the passenger exactly 65 ft above the ground? Solve algebraically; round each time to the nearest tenth of a second.
  3. During what fraction of each revolution is the passenger more than 65 ft above the ground? Justify using the times from (b) and the periodicity of \(h\).
  4. Suppose the wheel speeds up so that the new period is 60 seconds (no other parameters change). Write the new function \(\tilde h(t)\). Describe in one sentence the geometric difference between \(h\) and \(\tilde h\) using the language of horizontal stretch / compression — and state the compression factor.
10. Sequences, series, and annuities 9 points

A retirement contributor deposits $200 at the end of each month for 30 years (360 deposits). The account earns interest at a nominal annual rate of 6%, compounded monthly. Use the monthly growth factor \(r = 1 + 0.06/12 = 1.005\) throughout.

  1. Let \(a_n\) denote the value at the end of month 360 of the deposit made at the end of month \(n\), for \(n = 1, 2, \ldots, 360\). Write \(a_n\) as an explicit function of \(n\). State whether \((a_n)\) is arithmetic, geometric, or neither, and (if geometric) state the common ratio.
  2. Express the final account value \(S = a_1 + a_2 + \cdots + a_{360}\) as a finite geometric series. Apply the formula \( \dfrac{r^{N} - 1}{r - 1} \) to compute \(S\). Round to the nearest dollar.
  3. After year 30, the contributor stops depositing and lets the balance accumulate undisturbed for 5 more years at the same monthly rate. Find the balance at the end of year 35. Round to the nearest dollar.
  4. Compute the infinite geometric sum \( \displaystyle \sum_{k=1}^{\infty} 1000 \cdot \left(\tfrac{1}{2}\right)^{k-1} \) using the formula \( \sum_{k=1}^{\infty} a r^{k-1} = \dfrac{a}{1 - r} \) (valid when \(|r| < 1\)). Justify the convergence in one sentence.
11. Conic sections — parabolic suspension cable 8 points

The roadway of a small suspension bridge is a horizontal segment of length 200 ft. The supporting cable is a parabola whose lowest point is exactly 50 ft below the roadway and centered on it. The two ends of the cable just meet the roadway at the two ends of the bridge. Vertical suspender rods connect the roadway to the cable.

  1. Place a coordinate system with the lowest point of the cable at the origin and the roadway running horizontally above it. Write the equation of the parabolic cable in the form \( y = a x^2 \). Determine \(a\) exactly.
  2. Suspender rods are placed every 25 horizontal feet, beginning at the center. Compute the length of each rod (the vertical distance from cable to roadway) at horizontal positions \(x = 0,\,25,\,50,\,75,\,100\). Present the answer as a small table, lengths to the nearest foot.
    Horizontal position \(x\) (ft)Rod length (ft)
    0 
    25 
    50 
    75 
    100 
  3. Find the total length of all suspender rods. Count the central rod (\(x = 0\)) once and each pair at \(x = \pm 25,\,\pm 50,\,\pm 75\) twice; do not count rods at the ends, which have length 0. Show your sum.
  4. Find the equation of the parabola from (a) in vertex form \(y = a(x - h)^2 + k\). State the vertex and the direction of opening, and verify the equation passes through \((100, 50)\).
12. Statistical inference — proportion test & confidence interval 12 points

A school district claims that 70% of its graduates enroll in college within one year of graduation. A skeptical board member surveys a simple random sample of 250 recent graduates and finds that 162 of them enrolled in college within that window.

  1. Compute the sample proportion \(\hat p\) to four decimal places.
  2. Compute the standard error of \(\hat p\) using the formula \( \text{SE} = \sqrt{\hat p (1 - \hat p)/n} \). Round to four decimal places.
  3. Construct a 95% confidence interval for the true population proportion \(p\) using \(\hat p \pm 1.96 \cdot \text{SE}\). Give the interval to four decimal places.
  4. Write one full sentence interpreting the interval in context.
  5. Does the district's claim of 70% lie inside the interval? In one sentence, comment on whether the data is consistent with the district's claim.