Math 2 Midterm Review: Tests 2 – 5 Name:_______________________

Test #2 (Solving Quadratics Equations and Inequalities and Complex Numbers) How do you solve quadratic equations by factoring?

o put quadratic into standard form (y =ax2+bx+c) o Factor o set factors = 0 o solve

1. x2 - 12 = x 2. x2 + 7x + 4 = -2 3. 2x2 – 18 = 0

How do you solve quadratic equations with the quadratic formula?

o put into standard form o find a,b, and c o substitute into the quadratic formula (on your formula sheet) o simplify (if the radicand is negative take out i)

4. 6x2 + 11x – 10 = 0 5. x2 + 4x = -6 6. -2x2 – 10 = - 5x

How do you simplify expressions involving complex numbers?

o Treat i as a variable o i2 = -1

o = i o

7. 8. (3 + 2i) + (-5 – 9i) 9. (2 – i) – (3 – 5i)

10. (4 – 5i)2 11.

What does the discriminant tell you about quadratic equations?

o Discriminant: b2 – 4ac o If b2 – 4ac > 0 2 real roots o If b2 – 4ac = 0 1 real root o If b2 – 4ac < 0 2 complex (non-real roots)/no real roots

Determine the Nature of the Roots

12. x2 – x + 1 = 0 13. 4x – 2x2 + 7 = 0 14. 25x2 – 20x = -4

How do you graph inequalities on a number line? o Open circle < or > o Closed circle ≤ or ≥

How do solve quadratic inequalities? o Solve as an equation o Plot the two answers on a number line o Test the three intervals o Shade where its true o Write the solution as an inequality

15. Solve: x2 – 24 > -2x 16. 3x2 + 4x – 5 ≤ 0

Test #3 (vertex form, graphing transformations of quadratics, characteristics of

quadratics)

What does the graph of a quadratic function look like, and how do you describe its

characteristics?

o Looks like a parabola (u-shape)

o

Graph the quadratic function. Identify the vertex, x-intercept(s), y- intercept, the equation of the axis of symmetry, domain, range, zeros, which way it opens, and the interval of increase and decrease.

17. y = x2 – 2x – 8 18. y = -

x2 + 3

How do you convert a quadratic function from standard from to vertex form?

o Find h, h =

o Find k, substitute h into the original equation

o Find a same as a in original equation

o Substitute a, h, and k into vertex form (on formula sheet)

19. f(x) = -x2 + 6x + 1 (put in vertex form)

How do you convert a quadratic function from vertex from to standard form?

o Rewrite (x + h)2 as (x + h)(x + h)

o FOIL

o Multiply your answer from 2 by a

o Add k

20. f(x) = 3(x + 2)2 + 5 (put in standard form)

How do you graph quadratic functions using transformations?

Disclaimer: These rules only work if your equation is in vertex form, y = a(x – h)2 + k.

Transformation General Equation Rule Example

Vertical Shift y = x2 + k If k is positive: Up If k is negative: down

y = x2 + 4 up 4 y = x2 – 3 down 3

Horizontal Shift y = (x – h)2 If h is positive: right If h is negative: left

y = (x – 7)2 right 7 y = (x + 5)2 left 5

Reflection over the x-axis

y = -ax2 If a is negative: Reflect over x-axis

y = -x2

reflect over x-axis Dilation: Vertical Stretch/ Vertical Compression

y = ax2 > 1: Vertical stretch < 1: Vertical compression

y = 2x2 y = -4x2 Vertical stretch y = .2x2 y = -½ x2

Vertical compression

Determine the vertex of the given functions. Describe any transformations.

21. y = x2 + 6x + 5 22. y = -2x2 – 12x – 10

6

4

2

-2

-4

-6

-5 5

6

4

2

-2

-4

-6

-5 5

6

4

2

-2

-4

-6

-5 5

Test 4 (Absolute Value/Piecewise Functions)

How can I …

o Graph an absolute value and piecewise function?

o Find their characteristics?

o Solve absolute value equations and inequalities?

o Find the inverse of a function and its domain and range?

o Determine if a function is 1 to 1?

23. Graph then write it as a piecewise function: y = -.5|6x – 6| + 1

Describe the following characteristics of each. Vertex 24. 25.

Axis of symmetry

Domain

Range

Zeros/x-intercepts

y-intercept

Max/min value

Point(s) of discontinuity

Intervals of Increase/Decrease/Constant

Solve each absolute value equation or inequality. Isolate the abs. value first!

26. -2|x – 3|+2 = - 10 27. |x+6| + 1 < 10 28. |-2x – 2| > 18

Inverse Functions – 1 to 1 functions must pass the horizontal line test

Which functions below are 1 to 1? State the domain and range of the inverse of each.

29. 30.

Find the inverse of each.

31. y = 3x – 7 32. f(x) = 2(x – 4)2 – 3 33. f(x)= x3 + 27 34. y = 4/x

4

2

-2

-4

-5 5

Test 5 (Exponential Functions)

How can I …

o Apply the properties of exponents to simplify expressions?

o State the characteristics of exponential functions?

o Graph exponential functions and their transformations?

o Solve exponential equations and inequalities algebraically and graphically?

o Use exponential functions to model growth and decay, compound interest?

Simplify each:

35. (3x2y5)(-4xy7) 36.

37. (-5x3y – 5 z 0) 4

Describe the characteristic of the given exponential function: Domain 38.

Range

Asymptotes

Zeros/x-intercepts

y-intercept

Intervals of Increase/Decrease

End behavior

Describe the transformations. # Function Parent

Graph Reflect over x-axis?

Vertical dilation?

Reflect over the y-axis?

Horizontal shift?

Vertical shift?

39 f(x) = -7x - 1 40 f(x) = 9 x-2 + 3 41 f(x) = 5-x+4

Graph the functions.

42. f(x) = - 4x+1 43. f(x) = 2(4)x – 2 44. f(x) = 4 – x

Solve each.

45. 4 -3x+1 =

46.

4

2

-2

-4

-5 5

47. Use the graph to solve: -2(3x+1) = -6

Solve exponential growth and decay problems and calculate compound interest.

Growth and Decay Formula: Compound Interest:

48. A population of 100 frogs increases at an annual rate of 22%. How many frogs will

there be in 5 years?

49. The current value of my new 2013 Aston Martin Rapide is $226,850. It only depreciates at a rate of 4.6% a year. What will the value of my car be in the 4 years? 50. Daddy Warbucks just passed away and left you an inheritance of $250,000. The only stipulation is that it must remain invested for 10 years at an interest rate of 6.2% compounded daily. How much will you receive after the 10 year period?

-2

-4

-6

-8

-5 5