Honors Algebra 2: Practice Circles and Parabolas

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  1. State all the salient (important) features of the graph of the equation, then sketch the graph of the equation: \(y = -2x^2 - 4x + 1\)
    2. Solution

    parabola, opens down. Vertex: \((-1, 3);\) Focus: \( \displaystyle \left( -1, \frac{23}{8} \right); \) Directrix: \( \displaystyle y = \frac{25}{8};\) x-intercepts: \( \displaystyle \left( -1, \pm \frac{\sqrt{6}}{2}, 0 \right);\) y-intercept \((0, 1);\) domain: \( x \in \mathbb{R};\) Range: \(y \le 3;\) axis of symmetry: \(x = -1.\)

  2. State all the salient (important) features of the graph of the equation, then sketch the graph of the equation: \( \displaystyle y =9x^2 - 54x + 9y^2 + y + \frac{249}{4}\)
    3. Solution

    This is a circle with center \(3, 0\) and radius \( \displaystyle \frac{5\sqrt{3}}{6};\) there are no y-intercepts, and the x-intercepts are at \( \displaystyle \left( 3 \pm \frac{5 \sqrt{3}}{6}, 0\right).\)

  3. Given the circle whose equation is \( x^2 + 2x + y^2 - 4y + 1 = 0,\)
    1. find the coordinates of the endpoints of the vertical diameter.
      2. Solution

      \( (-1, 0)\) \((-1, 4)\)

    2. find the equation of the parabola, in standard form, whose vertex is the higher endpoint of the vertical diameter, and whose focus is the lower endpoint.
      3. Solution

      \( \displaystyle y = -\frac{1}{16}x^2 - \frac{x}{8} + \frac{63}{16}\)

  4. Give the equation in standard form of the circle that is tangent to both the x- and y-axes, has a radius of 7, and is in the third quadrant.
    5. Solution

    \( (x + 7)^2 + (y + 7)^2 = 49\)

  5. Give the equation in standard form of the parabola whose focus is at \( \displaystyle \left( \frac{1}{2}, -3\right)\) and whose directrix is the y-axis.
    6. Solution

    \( \displaystyle x = y^2 + 6y + \frac{37}{4}\)

  6. At $5.00 per ticket, 525 people will come to see a musical. For every dollar increase in ticket price, 25 fewer people buy tickets. What should the producer charge per ticket to make a maximum profit? What is the maximum profit she will make?
    7. Solution

    $13; $4,225