User:Melikamp/alg-trig

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Real Numbers[edit]

Recall that .

1. List 4 smallest elements of the set

2, 4, 6, 8

2. List 4 smallest elements of the set

0, 1, 2, 3

3. If , , and , find
4. Graph and write in a set builder notation the set

5. Graph and write in a set builder notation the set

6. Graph and write in a set builder notation the set

7. Evaluate given expressions if , , and

Rational Exponents[edit]

10. Evaluate

11. Evaluate

12. Simplify

13. Simplify

14. Simplify

15. Evaluate

16. Evaluate

17. Rationalize the denominator and simplify:

Polynomials[edit]

30. Rewrite the expression as a polynomial in the standard form and state the degree of the polynomial.

, degree 3

31. Rewrite the expression as a polynomial in the standard form and state the degree of the polynomial.

, degree 15

32. Rewrite the expression as a polynomial in the standard form and state the degree of the polynomial.

, degree 2

33. Evaluate the polynomial expression at

Factoring[edit]

We are only interested in factors that have integer coefficients.

40. Factor

41. Factor

42. Show that the polynomial is irreducible (that is, cannot be factored using real coefficients)

43. Factor

44. Factor

45. Factor

46. Factor

Linear Equations[edit]

50. You want to install hardwood floor tile. The delivery fee is $80, and the installation costs $6 per square foot (parts and labor). How many square feet can you tile on a $1100 budget?

170 square feet

51. Your distant relative's will stipulates that her money is to be divided between you, your sibling, and your child in such a way that your sibling gets twice as much as you do, and your child gets half as much as you do. How much money will you get if the total inheritance is seven hundred thousand dollars?

$200000

52. Solve the equation .

53. Solve the equation .

54. Determine whether the equation is conditional, a contradiction, or an identity.

Identity

55. Solve the equation .

56. Solve the equation .

Formulas[edit]

60. Solve the formula for (Mass-energy equivalence).

61. Solve the formula for (Fahrenheit to Celsius conversion).

Quadratic Equations[edit]

70. Solve the equation by factoring.

71. Solve the equation using the square root procedure.

72. Solve the equation by completing the square.

73. Solve the equation by using the quadratic formula.

Other Types of Equations[edit]

80. Solve the equation .

81. Solve the equation .

82. Solve the equation .

83. Solve the equation .

84. Solve the equation .

Functions and Graphs[edit]

90. Find the distance between points and .

13

91. Find the midpoint of the line segment connecting points and .

92. Find the intercepts of the graph of the equation .

x-intercept at , y-intercept at

93. Sketch the graph of the function and find its intercepts.

Polynomialdeg2.svg

94. Find the center and the radius of the circle by rewriting the equation in the standard form.

Center at , radius

Functions[edit]

100. Given that , find and .

and .

101. Find the domain of .

102. Find the domain of .

and .

103. Find the domain of .

104. Find the zeroes of .

105. Find the zeroes of .

106. Find the zeroes of .

Linear Functions[edit]

110. Find the slope of the line that passes through the points and .

111. Graph the function by finding the slope and the -intercept.
112. Graph the function given by by finding the slope and the -intercept.

In the next four exercises, state the answer in the form .

113. Find an equation for the line with the slope and containing the point .

114. Find an equation for the line containing the points and .

115. Find an equation of the line parallel to the line and containing the point .

116. Find an equation of the line perpendicular to the line and containing the point .

Quadratic Functions[edit]

120. Complete the square to find the standard form of the quadratic function and use it to sketch its graph.
121. Complete the square to find the standard form of the quadratic function and use it to sketch its graph.
122. Use the vertex formula to find the vertex of the graph of the quadratic function , and write the function in the standard form.
123. Use the vertex formula to find the vertex of the graph of the quadratic function , and write the function in the standard form.
124. Find the maximum or minimum value of the function and state the range of the function.
125. Find the maximum or minimum value of the function and state the range of the function.

Polynomial Long Division[edit]

130. Use polynomial long division to divide by .

131. Use polynomial long division to divide by .

Polynomial Functions of Higher Degrees[edit]

140. Use the Intermediate Value Theorem to show that the polynomial function Failed to parse (syntax error): {\displaystyle p(x) = x^6 − 3x^5 + 6x^2 − 1} has a zero in the interval .

141. Given that the function Failed to parse (syntax error): {\displaystyle \displaystyle f(x) = \frac{x^6 − 10}{x^5+10x^2−x}} is continuous on the interval , prove that it has a zero in that interval.

Zeros of Polynomial Functions[edit]

150. Find all zeroes of a polynomial function and state the multiplicity of each zero.

-4 (3), -3 (2), 3 (2)

151. Use the Descarte's Rule of Signs to state the possible numbers of positive and negative zeroes of the polynomial function .

3 or 1 positive zeroes, 1 negative zero.

152. Find the zeroes of the polynomial function and state their multiplicity.

(1), (1), (1)

153. Find the zeroes of the polynomial function and state their multiplicity.

(1), (1), (1), (1)

Fundamental Theorem of Algebra[edit]

160. Find a polynomial function of the lowest degree with roots .

161. Find all zeroes of a polynomial function , given that it has a root of multiplicity . State the answer by rewriting the polynomial as a product of linear factors.

Inverse Functions[edit]

170. Use composition of functions to determine whether and are inverses of each other.

Yes

171. Find the inverse of the function or prove that it does not exist.

This function is not injective, since

172. Given find the inverse of , and state the domains and the ranges for both and .

, ,

173. Given find the inverse of , and state the domains and the ranges for both and .

, ,

Exponential Functions[edit]

180. Evaluate for and .

and

181. Use a computer to estimate the value of for and .

and

182. Sketch the graph of the function .
183. Sketch the graph of the function .

Logarithmic Functions[edit]

190. Rewrite the equation in exponential form.

191. Rewrite the equation

192. Evaluate without using a computer.

193. Find the domain of the function .

194. Find the domain of the function .

Properties of Logarithms[edit]

200. Use a computer to estimate .

201. Rewrite the expression in a way that leaves the arguments of function as simple as possible.

202. Rewrite the expression as a single logarithm with coefficient .

Exponetial and Logarithmic Equations[edit]

210. Solve the equation .

211. Solve the equation .

212. Solve the equation .

213. Solve the equation .

Angles and Arcs[edit]

220. Find the complement and the supplement of the angle .

Complement is and supplement is

221. Find the complement and the supplement of the angle .

Complement is and supplement is

222. Convert the degree measure into the exact radian measure.

223. Convert the radian measure into the exact degree measure.

Right Triangle Trigonometry[edit]

230. Find the values of , , , , , and if in the right triangle with and side lengths and .

, , , , , and

231. Let be the acute angle in a right triangle and . Find and .

and

Trigonometric Functions of any Angle[edit]

240. Find the value of , , , , , and for the angle, in standard position, whose terminal side passes through the point .

, , , , , and

241. Find if and .

Trigonometric Functions of Real Numbers[edit]

250. Find the exact value of .

251. Find the exact value of .

252. Find the exact value of .

253. Rewrite in terms of a single trigonometric function or a constant.

254. Rewrite in terms of a single trigonometric function or a constant.

255. Rewrite in terms of a single trigonometric function or a constant.

Graphs of Sine and Cosine[edit]

260. Find the amplitude and the period of the function . Plot one full period.

Amplitude and period

261. Find the amplitude and the period of the function . Plot one full period.

Amplitude and period

262. Find the amplitude and the period of the function . Plot one full period.

Amplitude and period

263. Find the amplitude and the period of the function . Plot one full period.

Amplitude and period

Verification of Trigonometric Identities[edit]

270. Verify the identity .
271. Verify the identity .
272. Verify the identity .
273. Verify the identity .