Troy School District · Mathematics · Algebra 1 · Form B
Standards Drill · 120 minutes · 100 points

Algebra 1 — Standards-Aligned Skills Examination

A calculation-focused, standards-walkthrough exam covering every CCSS-M Algebra 1 conceptual category (N-Q, A-SSE, A-APR, A-CED, A-REI, F-IF, F-BF, F-LE, S-ID). Each problem header cites the standard(s). Form B complements Form A.

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Instructions

Part I · No Calculator · Problems 1–5 · 50 pts · ~50 min
1. A-REI.B · Linear equations and inequalities 10 points
  1. Solve each linear equation. Show every step:

    (i) \( 5x - 3 = 12 \)    (ii) \( 2(3x - 4) = 5x + 1 \)    (iii) \( \dfrac{x + 3}{2} - \dfrac{x - 1}{4} = 5 \)    (iv) \( -3(2x - 5) + 4 = 7 - x \)

  2. Solve each linear inequality. Graph each solution on a number line:

    (i) \( 4x - 7 \ge 17 \)    (ii) \( -2x + 9 < 1 \)    (iii) \( -3 \le 2x + 1 < 7 \)

  3. Solve each absolute-value equation/inequality:

    (i) \( |x - 5| = 8 \)    (ii) \( |2x + 3| < 7 \)    (iii) \( |3x - 4| \ge 11 \)

2. A-REI.C · Systems of linear equations 10 points
  1. Solve by substitution:

    (i) \(\begin{cases} y = 2x + 1 \\ 3x + y = 16 \end{cases}\)    (ii) \(\begin{cases} x = 4y - 3 \\ 2x + 3y = 16 \end{cases}\)

  2. Solve by elimination:

    (i) \(\begin{cases} 2x + 5y = 11 \\ 3x - 5y = 9 \end{cases}\)    (ii) \(\begin{cases} 4x + 3y = -1 \\ 6x - 5y = 21 \end{cases}\)    (iii) \(\begin{cases} x + 2y = 7 \\ 3x - y = 7 \end{cases}\)

  3. Determine without solving how many solutions each system has:

    (i) \(\begin{cases} 2x + 3y = 6 \\ 4x + 6y = 18 \end{cases}\)    (ii) \(\begin{cases} x - y = 4 \\ 3x - 3y = 12 \end{cases}\)

3. A-APR.A, A-SSE.A · Polynomial multiplication and factoring 12 points
  1. Multiply (FOIL or area method):

    (i) \( (3x - 2)(2x + 5) \)    (ii) \( (x + 4)^2 \)    (iii) \( (2x - 3)(2x + 3) \)    (iv) \( (x - 1)(x^2 + 2x + 3) \)

  2. Factor completely:

    (i) \( x^2 + 7x + 12 \)    (ii) \( x^2 - 11x + 30 \)    (iii) \( 2x^2 + 7x + 6 \)    (iv) \( 6x^2 - x - 12 \)    (v) \( x^2 - 81 \)    (vi) \( 9x^2 - 49 \)    (vii) \( 4x^2 + 12x + 9 \) (perfect square trinomial)    (viii) \( 5x^3 - 20x \) (factor a GCF first)

  3. Add or subtract polynomials, simplify:

    (i) \( (3x^2 - 4x + 5) + (2x^2 + 7x - 9) \)    (ii) \( (5x^2 - 3x + 2) - (2x^2 + 4x - 7) \)

4. A-REI.B.4 · Quadratic equations 12 points
  1. Solve by factoring:

    (i) \( x^2 - 9x + 14 = 0 \)    (ii) \( 3x^2 + 8x - 3 = 0 \)    (iii) \( x^2 = 81 \)    (iv) \( 4x^2 - 28x = 0 \)

  2. Solve by the quadratic formula. Express exactly (allow \(\sqrt{}\) and fractions):

    (i) \( x^2 - 6x + 7 = 0 \)    (ii) \( 2x^2 + 5x - 1 = 0 \)    (iii) \( 3x^2 - 4x + 5 = 0 \) (no real solutions; show via discriminant)

  3. Solve by completing the square: \( x^2 + 6x - 4 = 0 \). Show every step.
  4. Convert to vertex form by completing the square: \( y = x^2 - 8x + 11 \). State the vertex.
  5. Use the discriminant \(b^2 - 4ac\) to determine the number of real solutions of each:

    (i) \( x^2 + 4x + 4 = 0 \)    (ii) \( 2x^2 - 5x + 7 = 0 \)    (iii) \( x^2 - 8x + 7 = 0 \)

5. N-RN.A, A-SSE · Radicals and exponent rules 6 points
  1. Simplify each radical:

    (i) \( \sqrt{72} \)    (ii) \( \sqrt{50} + \sqrt{18} \)    (iii) \( \sqrt{8} \cdot \sqrt{6} \)    (iv) \( \dfrac{\sqrt{27}}{\sqrt{3}} \)

  2. Simplify, expressing answers with positive exponents only:

    (i) \( x^5 \cdot x^{-3} \)    (ii) \( (2x^3 y^{-2})^2 \)    (iii) \( \dfrac{12 a^4 b^{-1}}{4 a^{-2} b^{3}} \)

  3. Convert: \(81^{3/4} = ?\) and \(125^{2/3} = ?\) (no calculator).
Part II · Calculator Permitted · Problems 6–10 · 50 pts · ~70 min
6. F-IF.A–B · Function notation and key features 10 points
  1. For \( f(x) = 3x^2 - 5x + 2 \), compute \(f(0), f(1), f(-2),\) and \(f(t + 1)\) (this last one symbolically).
  2. For \( g(x) = x^2 - 8x + 12 \):

    (i) Find the vertex.
    (ii) Find the \(x\)-intercepts.
    (iii) State the axis of symmetry and the range.

  3. Compute the average rate of change of \( h(x) = x^2 - 4x \) on \([1, 5]\). Show your work.
  4. For the linear function \( f(x) = -\tfrac{2}{3} x + 4 \), find the \(x\)-intercept and \(y\)-intercept and sketch on a labeled coordinate plane.
7. F-LE · Linear and exponential modeling 10 points
  1. Write a linear model \(L(t)\) for each:

    (i) starts at $50, decreases by $4 per week
    (ii) passes through \((2, 7)\) and \((6, -1)\)
    (iii) has slope \(-\tfrac{1}{4}\) and \(y\)-intercept 6

  2. Write an exponential model \(E(t) = a \cdot b^t\) for each:

    (i) starts at 200, doubles every period
    (ii) starts at 800, decays at 5% per period
    (iii) starts at 1, grows by 12% per period

  3. For the model \( P(t) = 50 \cdot (1.08)^t \), find:

    (i) the initial value    (ii) the growth rate per period as a percent    (iii) \(P(10)\) (round to 2 decimals)

  4. For \( Q(t) = 100 \cdot (0.85)^t \), build a table of values for \(t = 1, 2, 3, 4, 5, 6\) and use it to estimate the smallest whole \(t\) at which \(Q(t)\) drops below 50.
8. F-BF · Function transformations 8 points
  1. Describe the transformation taking \(y = f(x)\) to:

    (i) \( y = f(x - 4) \)    (ii) \( y = f(x) + 2 \)    (iii) \( y = -f(x) \)    (iv) \( y = f(-x) \)    (v) \( y = 3 f(x) \)    (vi) \( y = f(2x) \)

  2. Starting from \(y = x^2\), write the equation of the parabola translated 3 units right and 5 units up, then reflected across the \(x\)-axis.
  3. The graph of \( g \) is obtained from \( f(x) = |x| \) by stretching vertically by factor 2 and shifting left 1. Write \(g(x)\).
9. S-ID · Univariate and bivariate statistics 12 points

Test scores: \(72, 78, 80, 85, 88, 90, 92, 95, 95, 100\).

  1. Compute the mean.
  2. Compute the median.
  3. Compute the mode.
  4. Find the five-number summary (min, Q1, median, Q3, max). Sketch a box plot.
  5. Compute the range and the IQR.
  6. Use the calculator's 1-Var Stats to compute the sample standard deviation \(s\). Round to two decimals.
  7. Bivariate: a line of best fit for the data points \( (1, 5), (2, 7), (3, 10), (4, 12), (5, 14) \) is given by \(\hat y = 2.30x + 2.70\). Plot the points and the line on a labeled coordinate plane. Then describe the direction (positive/negative) and strength (strong/weak) of the relationship by inspection.
10. A-CED, A-REI · Systems of inequalities 10 points
  1. Graph each linear inequality on a coordinate plane (use shading):

    (i) \( y \le 2x - 3 \)    (ii) \( y > -\tfrac{1}{2} x + 4 \)    (iii) \( x + y \ge 5 \)

  2. Graph the solution region of the system \[ \begin{cases} y \ge x - 2 \\ y \le -2x + 4 \\ x \ge 0 \end{cases} \] on a labeled coordinate plane. Shade the region and identify one specific point that satisfies all three inequalities.
  3. Verify your point from (b) by substituting its coordinates into all three inequalities. State which boundary lines, if any, your point lies on.