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

Algebra 2 — Testing-Out Examination

A student who passes this examination has demonstrated mastery of the Common Core Algebra 2 standards (CCSS-M conceptual categories N-CN, A-APR, A-REI, F-IF, F-BF, F-LE, F-TF, S-ID) and is eligible to advance to Pre-Calculus.

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Instructions

Part I · No Calculator · Problems 1–4 · 30 pts · ~40 min
1. Polynomial factoring & sign analysis 8 points

Let \( P(x) = x^3 - 4x^2 + x + 6 \).

  1. Show that \(x = -1\) is a zero of \(P(x)\). Either substitution or synthetic division is acceptable; show the work.
  2. Factor \(P(x)\) completely over the integers.
  3. Make a sign chart for \(P(x)\) and use it to write the solution set of \(P(x) > 0\) in interval notation.
  4. Sketch \(y = P(x)\) on a labeled coordinate plane. Mark each \(x\)-intercept and indicate the qualitative location of any local extrema (you do not need to compute them exactly).
2. Complex numbers & quadratic equations 7 points
  1. Simplify \( (3 - 2i)(4 + i) \). Write your answer in the form \(a + bi\).
  2. Simplify \( \dfrac{5 + i}{2 - 3i} \). Write your answer in the form \(a + bi\). Show the conjugate-multiplication step explicitly.
  3. Solve \( x^2 - 6x + 13 = 0 \) using the quadratic formula. Express your two solutions exactly and write each in the form \(a + bi\).
  4. A quadratic equation with real coefficients has \(2 - 3i\) as one solution. State the other solution and explain in one sentence which property of polynomials with real coefficients forces this.
3. Composition, inverse, and key features 7 points

Let \( f(x) = 2x - 5 \) and \( g(x) = \dfrac{x + 1}{x - 2} \).

  1. Compute \( (f \circ g)(3) \) and \( (g \circ f)(3) \). Show each substitution.
  2. Find \( f^{-1}(x) \) and \( g^{-1}(x) \). Show each step. State the domain of \(g\) and the domain of \(g^{-1}\).
  3. Verify directly that \( g\bigl(g^{-1}(0)\bigr) = 0 \) using your formula from (b).
4. Logarithm properties (no calculator) 8 points
  1. Without a calculator, evaluate

    (i) \( \log_2 32 \)     (ii) \( \log_5\!\sqrt{125} \)     (iii) \( \log_3\!\dfrac{1}{27} \).

  2. Express \( \log_b\!\left(\dfrac{x^2 \, y}{z^3}\right) \) as a sum/difference of simpler logarithms. Name each property you apply.
  3. Solve for \(x\): \( \log_3(x + 2) + \log_3(x) = 1 \). Show all algebraic steps. State which solution(s) are valid and why any extraneous solution must be discarded.
Part II · Calculator Permitted · Problems 5–11 · 70 pts · ~80 min
5. Quadratic modeling — projectile 10 points

A small rocket is launched straight up from a 6-ft platform with an initial vertical velocity of 88 ft/sec. Its height \(h\) (in feet) above the ground at time \(t\) (in seconds) is given by \[ h(t) = -16 t^2 + 88 t + 6. \]

  1. What is the height of the rocket at \(t = 0\)? Interpret this value in context.
  2. Find the time at which the rocket reaches its maximum height, and find that maximum height. Use the vertex formula and show your work.
  3. Solve \( h(t) = 0 \) using the quadratic formula. Round to the nearest hundredth of a second. Which root is the physically meaningful answer? Why is the other root rejected?
  4. Sketch \(y = h(t)\) for \(0 \le t \le 6\). Label the vertex, the y-intercept, and the time at which the rocket lands.
6. Rational function — asymptotes & key features 10 points

Let \( \displaystyle f(x) = \frac{x^2 - 9}{x^2 - 4x + 3} \).

  1. Factor numerator and denominator completely. Identify any holes (removable discontinuities) and any vertical asymptotes. State each as an \(x\)-value with a brief justification.
  2. Determine the horizontal asymptote of \(y = f(x)\). Justify by comparing degrees of numerator and denominator.
  3. Find the \(x\)-intercept(s) and \(y\)-intercept of \(y = f(x)\) (if any). Show your work.
  4. Sketch \(y = f(x)\), labeling all asymptotes (as dashed lines), holes (as small open circles), and intercepts.
7. Exponential modeling — coffee cooling 9 points

A cup of coffee is poured at 195 °F in a room of constant temperature 70 °F. Five minutes later, the coffee has cooled to 160 °F. Newton's Law of Cooling models the temperature as \[ T(t) = T_{\text{room}} + (T_0 - T_{\text{room}})\, e^{-kt}, \] where \(t\) is time in minutes after pouring.

  1. Identify \(T_{\text{room}}\) and \(T_0\) from the problem. Then use the 5-minute data point to determine \(k\) exactly (in terms of \(\ln\)).
  2. Use your model to predict the temperature 15 minutes after pouring. Round to the nearest tenth of a degree.
  3. How long after pouring will the coffee cool to 100 °F? Solve algebraically and round to the nearest tenth of a minute.
  4. As \(t\) becomes very large, what value does \(T(t)\) approach according to the model? Explain in one sentence what this represents physically.
8. Logarithmic scale — sound intensity 8 points

The sound level \(\beta\) in decibels (dB) of a sound of intensity \(I\) (in W/m²) is given by \[ \beta = 10 \log_{10}\!\left(\frac{I}{I_0}\right), \qquad I_0 = 10^{-12}\ \text{W/m}^2. \]

  1. A normal conversation has intensity \(I = 10^{-6}\) W/m². Compute its decibel level.
  2. A jackhammer at close range has intensity \(I = 10^{-2}\) W/m². Compute its decibel level.
  3. Find the ratio of the jackhammer's intensity to the conversation's intensity. (Express as a power of 10.)
  4. If sound A is 20 dB louder than sound B, by what factor is sound A's intensity greater than sound B's? Justify symbolically using the formula.
9. Trigonometry — unit circle & sinusoidal model 9 points
  1. Find the exact value of each (no calculator):

    (i) \( \sin(5\pi/6) \)     (ii) \( \cos(-\pi/3) \)     (iii) \( \tan(7\pi/4) \).

  2. The temperature \(T(t)\) (in °F) at a particular city, \(t\) hours after midnight on a typical summer day, is modeled by \[ T(t) = 78 - 12\,\cos\!\left(\frac{\pi}{12} \,t\right). \] State the maximum and minimum temperatures predicted by the model and the times of day at which they occur.
  3. Using the same model, find all times \(t \in [0, 24)\) at which the temperature is exactly 84 °F. Solve algebraically and round to the nearest tenth of an hour.
10. Sequences & series 9 points
  1. An arithmetic sequence has \(a_1 = 7\) and \(a_{20} = 102\). Find the common difference \(d\) and a closed form for \(a_n\). Then compute \(\sum_{n=1}^{20} a_n\).
  2. A geometric sequence has \(b_1 = 5\) and common ratio \(r = 2/3\). Compute the exact value of \(\sum_{n=1}^{8} b_n\). Then compute the exact value of the infinite sum \(\sum_{n=1}^{\infty} b_n\) and explain why it converges.
  3. A ball is dropped from 40 ft. After each bounce, it rebounds to 70% of its previous height. Find the total vertical distance the ball travels (sum of all upward and downward motions, treating the rebounds as a geometric series). Round to the nearest tenth of a foot.
11. Statistics — distributions and intervals 10 points

A school board member is reviewing district end-of-year math assessment scores to validate the claim that the scores are normally distributed with mean \(\mu = 200\) and standard deviation \(\sigma = 25\). Assume the claim is correct for the questions below.

  1. What proportion of students score between 175 and 225? Justify using the empirical (68-95-99.7) rule.
  2. What proportion of students score between 150 and 250? Justify using the empirical rule.
  3. What proportion of students score above 245? Standardize (compute a \(z\)-score) and use a normal table or calculator. Round to four decimal places.
  4. What proportion of students score below 165? Standardize and use a normal table or calculator. Round to four decimal places.
  5. The board member labels any score above 240 "exceeds expectations." What percent of students would be classified that way? Round to one decimal place.