Honors Algebra 2: Assignment 57: Unit Review

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

Calculator Inactive.

1. Complete the square: $$3x^2 - 5x + 2$$
2. Solution

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

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

$$\displaystyle -2x^2 + 12x - 10$$

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

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

7. Find all the salient features and sketch the graph of $$\displaystyle y = -\frac{1}{3}(x + 1)^2 + 6$$
8. 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.

9. Find all the salient features and sketch the graph of $$y = 2x^2 - 12x + 4$$
10. 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.

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

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

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

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

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

$$\displaystyle y = x^2 - 14x + 45$$

17. 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.
18. Solution

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

19. Solve over the complex numbers: $$x^2 = 2x - 4$$
20. Solution

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

21. 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?
22. Solution

$$\displaystyle 4840$$

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

$$\displaystyle x = \left\{0, 18\right\}$$

25. Write a quadratic function in standard form that has solutions $$x = \{4 \pm 3i\}$$
26. Solution

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

27. Solve over the complex numbers:$$\displaystyle \frac{2}{x} + \frac{1}{\sqrt{x}} = 6$$.
28. Solution

$$\displaystyle x = \frac{4}{9}$$

Calculator Active

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

$$\displaystyle y = -2x^2 + 6x - 4$$

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

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

5. 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 $$h_0.$$
6. Solution

$$\displaystyle v_0 = 20; h_0 = 37.5$$