# Honors Algebra 2: Practice Circles and Parabolas

• Class: Honors Algebra 2
• Author: Peter Atlas
• Text: Algebra and Trigonometry: Structure and Method, Brown

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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: $$x = -1.$$

3. State all the salient (important) features of the graph of the equation, then sketch the graph of the equation: $$\displaystyle y =9x^2 - 54x + 9y^2 + y + \frac{249}{4}$$
4. 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 $$\displaystyle \left( 3 \pm \frac{5 \sqrt{3}}{6}, 0\right).$$

5. 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

$$(-1, 0)$$ $$(-1, 4)$$

3. 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.
4. Solution

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

6. 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.
7. Solution

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

8. 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.
9. Solution

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

10. 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? 11. Solution$13; \$4,225