Honors Algebra 2: Assignment 57: Unit Review

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  1. Complete the square: \(3x^2 - 5x + 2\)
    2. Solution

    \( \displaystyle 3 \left( x - \frac{5}{6} \right)^2 - \frac{1}{12} \)

  2. Change the following into standard form: \(y = -2(x - 3)^2 + 8\)
    3. Solution

    \( \displaystyle -2x^2 + 12x - 10 \)

  3. Change the following into \((h, k)\) form: \(y = 6x^2 + 18x + 1\)
    4. Solution

    \( \displaystyle y = 6\left(x + \frac{3}{2} \right) ^2 - \frac{25}{2} \)

  4. Find all the salient features and sketch the graph of \( \displaystyle y = -\frac{1}{3}(x + 1)^2 + 6\)
    5. Solution

    Vertex: (-1, 6); Axis of symmetry: \(x = -1;\) y-intercept: \( \displaystyle \left(0, \frac{17}{3}\right);\) x-intercepts:\( \displaystyle \left(-1 \pm 3\sqrt{2}, 0 \right); \) Reflection of the y-intercept over the axis of symmetry: \( \displaystyle \left( -2, \frac{17}{3} \right) \). For the graph, use your calculator to check.

  5. Find all the salient features and sketch the graph of \(y = 2x^2 - 12x + 4\)
    6. Solution

    Vertex: (3, -14); Axis of symmetry: \(x = 3;\) y-intercept: \( \displaystyle (0, 4)\) x-intercepts:\( \displaystyle \left(3 \pm \sqrt{7}, 0 \right); \) Reflection of the y-intercept over the axis of symmetry: \( \displaystyle \left( 6, 4 \right) \). For the graph, use your calculator to check.

  6. Find all values of \(k\) so that the quadratic function \(kx^2 - 6x + 3k = 1\) has one real, rational, double root.
    7. Solution

    \( \displaystyle k = \frac{1 \pm \sqrt{109}}{6} \)

  7. Find the equation in standard form of the parabola that has x-intercepts at 4 and -3, and has a y-intercept of 2.
    8. Solution

    \( \displaystyle y = -\frac{1}{6}x^2 + \frac{1}{6}x + 2 \)

  8. Find the equation in standard form of the parabola that has vertex (7, -4) and contains the point (4, 5).
    9. Solution

    \( \displaystyle y = x^2 - 14x + 45 \)

  9. The sum of the roots of a quadratic equation is 8, and the product is 10. Find the equation if the orientation of the parabola is that it opens down, and is squished vertically by a factor of 3.
    10. Solution

    \( \displaystyle y = -\frac{1}{3}x^2 + \frac{8}{3}x - \frac{10}{3}\)

  10. Solve over the complex numbers: \(x^2 = 2x - 4\)
    11. Solution

    \( \displaystyle x = \left\{1 \pm i\sqrt{3} \right\} \)

  11. A producer knows that charging $12 per ticket will generate an audience of 400 people. For each 20-cent increase in the cost of a ticket, 8 fewer people will come. What is the maximum revenue the producer of the show can make?
    12. Solution

    \( \displaystyle $4840 \)

  12. Solve over the complex numbers: \(\displaystyle x(x - 2) - 3x\sqrt{x - 2} - 4x = 0\)
    13. Solution

    \( \displaystyle x = \left\{0, 18\right\} \)

  13. Write a quadratic function in standard form that has solutions \(x = \{4 \pm 3i\}\)
    14. Solution

    \( \displaystyle x^2 - 8x + 25 \) or any scalar multiple thereof.

  14. Solve over the complex numbers:\( \displaystyle \frac{2}{x} + \frac{1}{\sqrt{x}} = 6\).
    15. Solution

    \( \displaystyle x = \frac{4}{9} \)

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  1. Find the equation of the quadratic that passes through the points (-2, -24), (4, -12) and \(\displaystyle \left(\frac{3}{2}, \frac{1}{2} \right).\)
    2. Solution

    \( \displaystyle y = -2x^2 + 6x - 4 \)

  2. Convert the quadratic \(y = p^2x^2 - 6x + p\) into \((h, k)\) form.
    3. Solution

    \( \displaystyle y = p^2 \left( x - \frac{3}{p^2} \right)^2 + \frac{p^3 - 9}{p^2} \)

  3. On an alien planet, a projectile thrown upward would have a height \(h(t) = -8t^2 + v_0t + h_0\) where \(h\) is measured in feet, and \(t\) is measured in seconds. The projectile hits the ground at \(t = 3.75\) seconds. The maximum height the projectile reaches is 50 ft. Find \(v_0\) and \(h_0.\)
    4. Solution

    \( \displaystyle v_0 = 20; h_0 = 37.5 \)