Algebra & Trigonometry Honors Midterm Review 016 Solving Equations 1) Find all values of x that satisfy the equation, 5x 1 = x + 3 ) Solve the following by completing the square. Express your answer in simplest a + bi form: x 6x1 0 3) Solve for x: 4) Solve for x: 5) Solve algebraically for x: 3 x 1 1 x 6) Solve for x: 1
7) Express, in simplest radical form, the roots of the equation x 4x 5 8) Solve for all values of x 9 9 1, x 0, x x 9) Solve for x: 10) Solve the following system of equations: x y 5 x y 7 11) Solve algebraically for x: x 6 8 4 1) Solve the following equation: 3 x x x 9 18 0
13) Solve the system below: 14) 3
Radicals and Imaginary Numbers 1. Simplify:.. Find the product of and. 3. Expand the expression (3 5). 4. Simplify: 4 8 18 5. The expression is equivalent to ) 4) 6. Express the sum of and in simplest radical form, in terms of i. 7. When simplified, is equivalent to 1) 1 3) -1 ) i 4) -i 8. Expressed in simplest form, is equivalent to 4
9. What is the sum of and the conjugate of? 10. 11. 1. The product of and is ) 4) 13. Simplify: 14. Simplify: 15. Simplify: 16. Simplify the following expression: 5 x 15 x 3 5
Polynomials, Inequalities, and Rational Expressions 1. Simplify:. What is the solution of the inequality? 3. What is the solution set of the inequality? 1) ) 3) 4) 4. Solve AND graph each inequality. Write your answer using interval notation. 5. Factor: 6. Factor: (x + 3) + (x + 3) 6 7. Factor: 6
8. Perform the indicated operation and express in simplest form: 4x 8 x x 1 3x 15 x x 4 8x 10 9. Express in simplest form: 10. Solve AND graph each inequality. Write your answer using interval notation. 10x + 9 > 5 11. Simplify for all values of x for which the expression is defined: 1. 13. Write an equation of a parabola with a focus at (3, 0) and a directrix with the equation of x = -3. 7
Exponents, and Logarithms 3 x 1. If f ( x) 1, then f 4 1) 8 3) equal to ) 4). Find the value of. Show your work. 3. Simplify the expression and write your answer using a positive exponent. 4. Simplify the following expression. Your answer should contain only positive exponents: 5. The expression, is equivalent to ) 4) 1 6. Which expression is equivalent to 1/ ab ) 4)? 8
7. Matt places $1,00 in an investment account earning an annual rate of 6.5%, compounded continuously. Using rt the formula V Pe, where V is the value of the account in t years, P is the principal initially invested, e is the base of a natural logarithm, and r is the rate of interest, determine the amount of money, to the nearest cent, that Matt will have in the account after 10 years. 8. The current population of Little Pond, New York, is 0,000. The population is decreasing, as represented by the 0.34 t formula P A(1.3), where population,, in years, and population. a. What will the population be 3 years from now? Round your answer to the nearest hundred people. b. To the nearest tenth of a year, how many years will it take for the population to reach half the present population? 9. Write an equation for a bank account which starts with $5,000 and gets 6.% interest compounded: a. Annually (each year): b. Monthly (1 times per year): c. Continuously: 10. Evaluate to the nearest tenth. 11. 9
1. is equal to ) 4) 13. The expression is equivalent to ) 4) 14. Solve for x: ln x x ln8 e lne e 15. 16. Which number satisfies 10
1. The roots of the equation are Roots, Proportionality, and Circles 1) Imaginary 3) real, irrational, and unequal ) real, rational, and equal 4) real, rational, and unequal. Which equation has imaginary roots? ) 4) 3. What are the sum (S) and product (P) of the roots of the equation? ) 4) 4. For which equation does the sum of the roots equal 3 and the product of the roots equal -4.5? 1) ) 3) 4) 5.. 6. If R is inversely proportional to A, and when, what is the value of R when? 1) 0.65 ) 1.6 3) 10 4) 6,50 11
7. Write an equation of a circle with a center that is (3, ) and passes through the point ( 5,8). 8. Write an equation of a circle that is tangent to the x-axis, with a center that is(,4). 9. Write the equation of the following circle: 10. Write the equation of the following circle: 11. Find the center and the radius of x y x y 8 10 37 0 1
Functions 1. Why is y x a function, but x y 9 not a function?. The accompanying graph shows the curves of best fit for data points comparing temperature to altitude in four different regions, represented by the relations A, B, C, and D. Which relation is not a function? 1) A ) B 3) C 4) D 3. What is the domain of the function f(x) = x + 4 16 x over the set of real numbers? 4. Write the following in function form: log y x 5. If and, find. 6. Given: and, find. 7. f () x x a) Determine if the function is one-to-one. b) Determine if the function is onto for the set of real numbers. c) Determine if the function is onto for the positive real numbers. 8. f x x x x 3 ( ) 5 6 a) Determine if the function is one-to-one. b) Determine if the function is onto for the set of real numbers. 13
9. Given the relation Which statement is true? 1) Both A and are functions. 3) Only A is a function. ) Neither A nor is a function. 4) Only is a function. 10. What is the inverse of the function? 1) ) 3) 4) 11. 1 f ( x) x 6. Find 3 f 1 ( x) and use composition to justify your answer. 1. What are the roots of this polynomial function? 13. 14