Ron Paul Curriculum 9th Grade Mathematics Problem Set #2

Transcription

1 Problem Set # Find the value of each expression below, and then name the sets of numbers to which each value belongs: (.6) / Name the property illustrated by each equation: 6. z + ( z ) = 7. (x + y) = x + 6y 8. = 9. 6() = (6). (m + n) = m + n

2 Problem Set # Do the following problem sets from the Khan Academy. Learn Math Algebra I Introduction to algebra Evaluating expressions in variables Combining like terms Combining like terms with distribution Simplify the following expressions by combining like terms:. y + y + y. v v + v + 8v. a + b 8 + a b +. Evaluate the expression in problem () using your graphing calculator for y = Evaluate the expression in problem () using your graphing calculator for v =. 6. Evaluate the expression in problem () using your graphing calculator for a = and b = 6.

3 Problem Set # Do the following problem sets from the Khan Academy. Learn Math Algebra I Solving Equations Two-step equations Two-step equations word problems Equations with variables on both sides

4 Problem Set #5 There is no problem set for Lesson #5.

5 Problem Set #6 Do the following problem sets from the Khan Academy. Learn Math Algebra I Solving inequalities Two-step inequality word problems Multi-step linear inequalities Compound inequalities

6 Problem Set #7 Solve the following absolute value equations. Check for extraneous solutions.. x =. y + 5 = y +. 9 t = + t. 8a = 6a Solve the following absolute value inequalities. Then graph the solutions on a number line. 5. n 6. 7 r < 9 7. m + 5> 7 8. Solve the following inequality for x in terms of a, b and c. Assume a, b, and c are real numbers and that a < and c >. ax + b c 9. The depth finder on a fishing boat gives readings that are within 5% of the actual water depth. When the depth finder reading is 5 feet, the actual water depth x lies in a range given by the following inequality: x 5.5x Write the absolute value inequality as a compound inequality, and then solve each part of the compound inequality for x. What are the possible actual water depths if the finder s reading is 5 feet?

7 Problem Set #8 Do the following problem sets from the Khan Academy. Learn Math Algebra I Functions Introduction to the domain and range of a function Domain and range from graph

8 Problem Set #9 Do the following problem sets from the Khan Academy. Learn Math Algebra I Functions Recognizing functions Recognize functions from graphs. Let f(x) = x x 6. a) Find f(). b) Find all x such that f(x) =. c) Find f(x ). d) Is there a real value of x such that f(x) = 6? Find the domain and range of each of the following real-valued functions. Write these in interval notation and in set builder notation.. f(x) = x. f(t) = t. h(x) = x 5. g(u) = + u 6. Let T(a, b, c) = a b c. a) Find T(,, 5) b) Find all values of x such that T(x,, 6) =.

9 Problem Set # Note: CLEP problems are multiple choice. Most problems in review problem sets will be CLEP style problems, but they will usually not be multiple choice, which I believe will be better practice.. Solve the inequality 9 x < and write the answer in interval notation.. Solve the equation x x = 9 for x.. In order to solve the equation 9 x + x=, you need to solve which of the following equations: (A) x + = x + (B) x 9x = (C) x 5x = (D) 77x x + 5 = (E) x + x =. Simplify the following expression: 5x(x x ) (x + x ) (x + ) 5. Which of the following expressions are irrational: I. 8 8 II. ( 8 ) III... (A) I only (B) II only (C) III only (D) I and II only (E) I and III only 6. What is the domain of x+ y =? x + 7. Solve for x: x 7.

10 8. The following series of algebraic steps simplifies the expression [5 + ( + ) + ]. It uses three algebraic laws: associative, commutative, and distributive. What is the order of the laws that are used as the expression is simplified in the following steps? [5 + ( + ) + ] = [5 + ( ) + ] = [5 + + ( + )] = (5) + () = + = (A) (B) (C) (D) (E) Distributive, Associative, Commutative Associative, Distributive, Commutative Commutative, Distributive, Associative Associative, Commutative, Distributive Commutative, Associative, Distributive 9. For how many integer values of x is x + x + = 6? (A) (B) (C) 5 (D) 7 (E) 9. What is the range of the function y = x +?

11 Problem Set # Do the following problem sets from the Khan Academy. Learn Math Algebra I Functions Inputs and outputs of a function Generate input-output pairs of a function Match function inputs to outputs from a graph Learn Math Algebra I Functions Piecewise functions Graphs of piecewise linear functions For each of the functions below, graph the function using () pencil and graph paper; () an online graphing program (for example, () your graphing calculator. Find the x-intercepts and the y-intercepts of the graph.. f (x) = (x ). f (x) = x + +. f (x) = 5. f (x) = x

12 Problem Set # Do the following problem sets from the Khan Academy. Learn Math Algebra I Two-variable linear equations Slope intercept equation from a graph Slope intercept equation from two solutions Point slope form Converting between slope intercept and standard form Find the slope m of a line perpendicular to each of the following lines:. x + y = 6. y = x + 9. y = 5x + 7

13 Problem Set # Do the following problem sets from the Khan Academy. Learn Math Probability and Statistics Regression Eyeballing the line of best fit Estimating slope of line of best fit

14 Problem Set # Do the following problem sets from the Khan Academy. Learn Math Algebra II Functions and their graphs Shift functions

15 Problem Set #5. Find the equation of the line passing through the points (, ) and (, ).. Which the following lines has the steepest slope? (A) y = (B) x + y = (C) x + y = (D) x y = (E) x + y = 7. What is the x-intercept of the graph of the equation y x =? 5. Mike goes into a pet shop with $. He wants to buy fish for his aquarium. Goldfish cost $.5 and Danios cost $.5 each. If he buys as many fish as possible with his money and buys at least one of each type of fish, what is the difference between the maximum number of fish he can buy and the minimum? (A) (B) (C) (D) (E) 5. Line k passes through the points (, ) and (6, ). Line l is perpendicular to line k with y-intercept 6. Find the x-intercept of line l. 6. Determine the values of M and N so that the graph of Mx + Ny = 5 will contain the points (, ) and (, ). (A) M =, N = (B) M =, N = 8 (C) M =, N = (D) M =, N = (E) M =, N =

16 7. Using the figure below, find the equation of the line perpendicular to the given line and passing through the given point. y x 8. Two inequalities are solved. The first one has the solution x <, and the other has the solution (, ]. If both inequalities must be true at the same time and x is an integer, how many possible solutions are there? (A) 5 (B) (C) (D) (E) 9. Using the figure in Problem #7 above, find an equation of the line that is parallel to the given line and passing through the point.. The greatest rational number among the following five choices is: (A) 5 (B) (5/) (C) 6. 5 (D) / (E) ( π)

17 Problem Set #6 Do the following problem sets from the Khan Academy. Learn Math Algebra II Advanced functions Compare features of functions

18 Problem Set #7 Do the following problem sets from the Khan Academy. Learn Math Algebra II Polynomial equations, expressions and functions End behavior of polynomial functions Determine the end behavior of polynomials Learn Math Algebra II Polynomial equations, expressions and functions Introduction to symmetry of functions Determine if a function is even or odd from its graph For the following functions, find f + g, f g, fg, and f/g and their domains.. f(x) = x, g(x) = x. f(x) = x +, g(x) = x. f(x) = x, g(x) = + x. f(x) =, g(x) = x x +

19 Problem Set #8 Find the functions f o g, go f, f o f, and go g and find their domains.. f(x) = x, g(x) = x +. f(x) = /x, g(x) = x +. f(x) = x, g(x) = x + Find the functions f o go h and find their domains.. f(x) = x, g(x) = x, h(x) = x 5. f(x) = x +, g(x) = x 5, h(x) = x Find functions f and g such that the following functions equal f o g. 6. F(x) = (x 9) 7. G(x) = x x + Do the following problem sets from the Khan Academy. Learn Math Algebra II Manipulating functions Evaluate composite functions from formulas Evaluate composite functions from graphs and tables Model situations by composing functions

20 Problem Set #9 Do the following problem sets from the Khan Academy. Learn Math Algebra II Functions Evaluate inverse functions Find inverse functions

21 Problem Set #. If f ( x) = and x g ( x) x =, what values of x are NOT in the domain of f(g(x))?. If f (x) = x x + and g(x) = x, what is the value of f (g(5))?. Which of the following graphs is not a function? (A) y (B) y x x (C) y (D) y x x (E) y x

22 . Two functions f (x) and g(x) are defined at x = by the table below. f (x) is an even function and g(x) is an odd function. A third function h(x) is defined by f (x) + g(x). What is the value of h( )? x f (x) g(x) 5 5. If f ( x) = x 8 and f ( ) x expression for f ( x )? denotes the inverse function of f, what is the 6. In the figure below, f (x) = x and g(x) is a transformation of f (x). Which of the following is the equation of g? (A) g(x) = x (B) g(x) = (x + ) g(x) y f (x) (C) g(x) = (x ) x (D) g(x) = (x + ) (E) g(x) = (x ) 7. The equation of the graph below could be: (A) x y = (B) y = x + x (C) x = y + y (D) x = y y y (E) y = x y x If f (x) = x, find f (a + ) f (a ).

23 9. The graphs of f(x) and g(x) are shown below. Find the value of g(f ( )) f (g( )). g(x) f (x). Which of the following graphs has both of these characteristics? (I) It is a function. (II) Its inverse is a function. (A) y (B) y x x (C) y (D) y x x (E) y x

24 Problem Set # Do the following problem sets from the Khan Academy. Learn Math Algebra II Complex Numbers Powers of the imaginary unit Classify complex numbers Adding and subtracting complex numbers Multiplying complex numbers Simplify the following expressions: m n. ( )( + ) Solve each equation:. 5x = 5. 7x + 8 = Find the values of m and n that make each equation true: i = m + ni 7. (m + n) + (m n)i = 7 i 8. (m n)i + (m + n) = + 7i 9. Find all real values of m such that the product ( + 5i)( + mi) is a real number.. Let z = a + bi and let z = a bi. Prove that z z is real.

25 Problem Set # Do the following problem sets from the Khan Academy. Learn Math Algebra II Complex Numbers Plot numbers on the complex plane Learn Math Precalculus Complex Numbers Divide complex numbers. Let z = a + bi and w = c + di, where a, b, c and d are real numbers. a) Prove that z + w = z + w. b) Prove that zw = z w.. Prove that z = z for all complex numbers z.. Prove that z = z if and only if z is real.. Prove that z = z if and only if z is pure imaginary.

26 Problem Set # Do the following problem sets from the Khan Academy. Learn Math Precalculus Complex Numbers Absolute value of complex numbers. Prove that Re(z) = z + z and that Im(z) = z z i for all complex numbers z.

27 Problem Set # Graph each of the following equations on the complex plane:. z + z =. z = z + i. ( i)z + ( i)z = 6i. z + i = 6 5. z i = z + i

28 Problem Set #5. The number fits into which classifications? I. Complex II. Real III. Rational. (A) (B) (C) (D) (E) + i i I only II only III only I and II only I, II and III is equivalent to: (A) + i (B) + i (C) i (D) (E). Let z = + i. If z is the complex conjugate of z, which of the following points shown on the figure below represents z z graphed in the complex plane? y B D E x A C. Suppose a = /. Which of the following is NOT a real number?

29 (A) a (B) a (C) a (D) a / (E) a 5. When + i + i is expressed in the form a + bi, what is the value of a? 6. ( + i) ( i) = (A) (B) (C) (D) (E) 7 9i 7 + 5i 7 + i 7 i 7 + i 7. Which of the points in the figure below represents the complex number graphed on the complex plane? y E + i i D x C B A 8. If i =, then (a + bi) (a bi) is equivalent to: (A) abi (B) (C) a b (D) bi (E) b

30 9. Simplify 5.. If x = + i and y = + i, find y x.

31 Problem Set #6 Do the following problem sets from the Khan Academy. Learn Math Algebra II Systems of equations and inequalities Graphing systems of equations

32 Problem Set #7 Do the following problem sets from the Khan Academy. Learn Math Algebra II Systems of equations and inequalities Systems of equations with elimination Systems of equations with substitution Systems of equations word problems Systems with one, zero or infinite solutions

33 Problem Set #8 There are no problems for lesson 8. If you like, find a website with an interactive D equation or function grapher and play around with. Try to experiment with as many different kinds of mathematical operators as you can remember, and see if you can guess what the D graph will look like before you graph it.

34 Problem Set #9 Find all the solutions of the systems of equations below. If the system has infinitely many solutions, write the solution set in terms of a parameter t, so that z = t.. x y z = y + z = x + y z = 5. y z = x + y = x y + z =. x + y z = x + y = x + y + z =. x + y z = x + y z = x + y z = 5. x + y z = x + y z = x y + z = Solve the following linear system with four variables using the same techniques that we learned for systems with three variables. It may be helpful to rewrite the system so that the variables are in columns (line up vertically). 6. x + z + w = 6 y z = x + y z = x + y + z w =

35 Problem Set # Do the following problem sets from the Khan Academy. Learn Math Algebra II Matrices Matrix dimensions Representing relationships with matrices

36 Problem Set # Note: For this problem set, solve the problem first on paper, and then check your answer with your graphing calculator. Do the following problem sets from the Khan Academy. Learn Math Algebra II Matrices Scalar matrix multiplication Matrix addition and subtraction Multiplying a matrix by a matrix Defined and undefined matrix operations Find the following matrix products:. = 6 6. = Let: Find AB and BA. = A [ ] B=

37 Problem Set # When necessary, use the following preimage to answer the questions below. The transformation matrices are defined below. Translation by a distance T x in the x direction and T y in the y direction: add Reflections: R x =, R y =, R y=x =, R y= x = Rotations: R 9 =, R 8 =, R 7 =, R 6 =. Find the matrix for the image after a translation to the right four units and down four units followed by a rotation of 7º.. Show that a rotation through 9 followed by a rotation through 8 is equivalent to a rotation through 7, and show that the order of the rotations does not matter. y x y y x x T T T T T T L

38 . Show that a reflection across the y-axis followed by a reflection across the line y = x is equivalent to a rotation by 7, and that it is not equivalent to a reflection across the line y = x followed by a reflection across the y-axis.. When a geometric figure is made larger or smaller we say that it is scaled. Find: a) a matrix S x that scales all of the x-coordinates by a factor of k b) a matrix S y that scales all of the y-coordinates by a factor of k c) a matrix S k that scales both x- and y-coordinates by a factor of k d) a matrix S a, b that scales the x-coordinates by a factor of a and the y-coordinates by a factor of b 5. Use the results from problem above to scale the preimage represented by the matrix 8 by a scale factor of ½. Then graph both the preimage and the image in the coordinate plane. 8

39 Problem Set # Solve the following systems of linear equations using augmented matrices.. x y + z = y + z = 5 x + y + z = 8. x + x x = 9 x x = x + 5x + x =. x + y + z w = y + z + w = x + y + w = x + z + w = 5. x + y + z w = x y + z + w = 9 x y + z + w = x + y + z w = Determine whether each of the following systems of linear equations is inconsistent or dependent. If it is dependent, find the complete parameterized solution. 5. x + y + z = y z = x + y + 5z = 6. x y 9z = 5 x + z = x + y z = 7. x y + z = x 8y + z = x y + z = 8. x + y z = x.5y + z = 6 x +.5y z = 8

40 Problem Set #6 Find the inverses of each of the following matrices, if it exists. Check your solutions with your graphing calculator

41 Problem Set #7 Solve the following systems of equations by writing them as matrix equations and using the inverse of the coefficient matrix. Note: You found some of the inverses needed in Problem Set #6.. x + 5y = 5x y =. 5x + y = x + y =. x + y + z = 7 x + y z = x + y =. y + z = x + y + z = x y + z = 8 5. x + y + w = y + z + w = y + w = x + y + w =

42 Problem Set #8 Find the determinants of each of the following matrices Find the following minors and cofactors using the matrix A. = 5.5 A. M, A 5. M, A 6. M, A 7. M, A 8. Find the determinant of the following matrix by expanding the determinant about: (a) The first row (b) The second column (c) The third row

43 Problem Set #9 Find the determinants of each of the following matrices, using row or column transformations whenever possible to simplify your work. Check your answers with your graphing calculator Find the determinants of the following matrices using the method of diagonals.. = A 5. = 8 7 B 6. Use the matrix at right to find the determinant by: (a) The method of diagonals (b) The method of cofactors (expand about the nd column) and then show that the results are equal. = i h g f e d c b a A