Exam 2 Review Chapter 1 Section1 Do You Know: 1. What does it mean to solve an equation? To solve an equation is to find the solution set, that is, to find the set of all elements in the domain of the variable that make the equation true. 2. What the difference is between an equation and an expression? An equation is a mathematical statement that two algebraic expressions are equal. Thus, the difference between the two is the equal sign. 3. How to tell if an equation is linear? An equation is linear if it can be written, through simplification, in the form. 4. How can you check your solution to an equation? To check a solution, substitute the value obtained into the original equation to see whether a true statement is obtained. 5. How do you check your solution to a word problem? Need to check and see if the answer makes sense in the original problem, not just the equation you wrote. Practice Problems: 1. Solve the equation: 2. Solve the equation: 3. Solve for c
4. A new game show requires a playing field with a perimeter of 54 yard and length 3 yards less than twice the width. What are the dimensions of the field? 5. A two-women rowing team can row 1,200 meters with the current in a river in the same amount of time it takes them to row 1,000 meters against that same current. In each case, their average rowing speed without the effect of the current is 3 meters per second. Find the speed of the current.
Chapter 1, Section 2 Do You Know: 1. What the difference is between solving linear equations and linear inequalities? The sense of an inequality reverses if we multiply or divide both sides by a negative number. There is no corresponding distinction in solving an equation. 2. What is the Union? The union of sets A and B, denoted, is the set formed by combining all the elements of A and all the elements of B. 3. What is the intersection? The intersection of sets A and B, denoted by, is the set of elements of A that are also in B. Practice Problems: 1. Write in inequality notation: 2. Write in interval notation: 3. Write in interval notation: 4. Write in inequality notation 5. Write in inequality and interval notation. -8-4 0 2 6. Solve and graph. 7. Solve and graph. -1 0
0 12 8. Solve and graph. -2 0 3 9. Graph the indicated set and write as a single interval if possible. -5 0 4 5 7 10. Graph the indicated set and write as a single interval if possible. 0 1 6 9 11. Solve and graph.
-5 0 15 12. For what real number(s) x does the expression represent a real number? Chapter 1, Section 3 1. What is the definition of absolute value? Absolute value is the distance from a point to the origin. 2. How to use absolute values to solve radical inequalities? See problems 5 and 6 below. Practice Problems: 1. Write without absolute value signs. 2. Solve the equation. 3. Solve the inequality. Write solutions to inequalities using both inequality and interval notation.
4. Solve the inequality. Write solutions to inequalities using both inequality and interval notation. 5. Solve the inequality. Write solutions to inequalities using both inequality and interval notation. 6. Solve the inequality. Write solutions to inequalities using both inequality and interval notation. Chapter 1, Section 4 1. What the standard form is for a complex number? 2. Do negative numbers have square roots? Explain. In the complex number system, EVERY real number has an (imaginary) root. 3. Is it possible to square an imaginary number and get a real number? Explain. Yes, the square of any pure imaginary number is a negative real number. 4. What is the conjugate of a complex number? The conjugate of is, when the two are multiplied together we get a real number.
5. How do we use conjugates? We use conjugates to get our imaginary number into standard form. 6. Which statement is false, and which is true? Justify your response. (A) Every real number is a complex number. True. Every real number can be written as a complex number (B) Every complex number is a real number. False. For example, is a complex number that is not a real number. 7. Is it possible to add a real number and an imaginary number? If so, what kind of number is the result? Yes, we can add a real number with an imaginary number, and we will get an imaginary number back. Practice Problems: 1. For each number find the (A) real part, (B) imaginary part, and (C) conjugate. (a) (b) (c) (d) 2. For the following, perform the indicated operation and write your answer in standard form. (a) (b) (c) (d) = 3. For the following, evaluate and express results in standard form. (a) (b) (c) (d) 4. Solve for x and y. (a)
(b) (c) 5. Solve for z and write your answer in standard form. (a) (b)
Chapter 1, Section 5 1. How can you tell when an equation is quadratic? That is, what is standard form for a quadratic equation? 2. What do a, b, and c in the quadratic formula stand for? They are constant coefficients where, x is a variable that changes. 3. How to explain the zero product property? If the product of two factors is zero, then at least one of those factors has to be zero. 4. How to explain the square root property? If 5. What is the Quadratic Formula? 6. How to explain the process of completing the square? That is, what are the steps to completing the square? 1. Get rid of the lead coefficient. 2. Get loose terms to the other side. 3. Take half the middle term, square it, and add to both sides. 3. Factor left side. 4. Solve for x. 7. What does the discriminate tell us about a quadratic equation? It tells us how many roots the quadratic has. Practice Problems (Remember with these problems you CAN have complex numbers as a solution): 1. Solve by factoring. 2. Solve by using the square root property.
3. Use the discriminant to determine the number of real roots of each equation, then solve using the quadratic formula. 4. Solve by completing the square. 5. Solve by completing the square.
6. Solve the following by any method. (a) (b) (c)
Chapter 1, Section 6 1. What is meant by the term extraneous solution? If an equation is solved by raising both sides to the same power, the resulting equation may have solutions that are not solutions of the original equation; these are called extraneous solutions. 2. When is it necessary to check for extraneous solutions? When we square both sides of an equation. 3. How can squaring both sides help in solving absolute value equations? Since, an absolute value equation can be regarded as a radical equation/ these can often be solved by squaring both sides. 4. How can you recognize when an equation is of quadratic type? If it has the form Practice Problems: 1. Solve the following equations (Remember you CAN have answers which are complex numbers). (a) (b) (c)
(d) Check the solutions in the original equation!!! (e) (follow example 4 on page 101) (f) (Follow example 3 on page 100) (g) (Follow example 1B on page 98) 2. A hard court version of the game broomball becomes popular on college campuses because it enables people to hit each other with a stick. The court is rectangle with diagonal 40 feet and area 800 square feet. Find the dimensions to one decimal place. (Hint: Use the Pythagorean Theorem) See example 5 on page 102 Chapter 2, Section 1 1. How to explain how to graph an equation of two variables using point-by-point plotting. To each Point in the plane there corresponds a single ordered pair of numbers (a,b) called coordinates of the point. To each ordered pair of number (a,b) there corresponds a single point, called the graph of the pair. 2. How to tell if a graph is symmetric about the y-axis, x-axis, and/or origin?
See Theorem 1 (Test for Symmetry) on page 115 Practice Problems: 1. Plot the given points in a rectangular coordinate system. (a) Now reflect the points (b) Reflect the points (c) Reflect the points over the y-axis. over the x-axis over the origin. Follow Example 3 on Page 114 for the Problem 1. 2. Test for symmetry with respect to the x-axis, y-axis, and the origin. (a) (b) (c) (d) Symmetric about the origin Symmetric about the y axis Symmetric about the origin Symmetric about the x-axis, y-axis, and origin Follow Example 4 on page 115 for problem 2. Chapter 2, Section 2 1. What is the Pythagorean Theorem? 2. How do can you calculate the distance between two points in the plane if you know their coordinates? 3. How can you calculate the midpoint of a line segment if you know the coordinates of the endpoints? 4. What is the standard form of a circle? 5. Can you explain how to find the standard form of the equation of the circle with center (1,5) and radius? See Example 6 on page 128 Practice Problems: 1. Find the distance and the midpoint of the line segment with end points 2. Find the indicated point when M is the midpoint and A and B are endpoints.
3. Find the center and radius of the circle. 4. Find the standard form of the equation of the circle knowing Center: (0,5), and point on circle: (2,-4). Chapter 2, Section 3 1. What is the standard form for a line? 2. How to explain how to find the x and y intercepts of a line if the equation is written in standard form? The x intercept is found by setting y=0, and the y intercept is found by setting x=0 3. What is slope-intercept form of a line? Why do you think it is called slope-intercept form? where 4. What is point-slope form of a line? Why do you think it is called point slope form? 5. How do you know if two lines are parallel? How do you know if they are perpendicular? Parallel if they have the same slope. Perpendicular if the slopes are negative reciprocals of each other. 6. What is the slope of the line? How do you find it? Change in y over change in x. 7. What is the slope of a horizontal line? m=0
8. What is the slope of a vertical line? m=undefined Practice problems: 1. Graph the equation and indicate the slope, if it exists. See Example one page 133 2. Find an equation for a line with slope= -3 and y intercept=7, write your answer in standard form. See Example 4A on page 137 3. Find an equation for a line which passes through the point (-5,4) and m=, write your final answer in slope-intercept form. See Example 5A on page 139 4. Given the x-intercept of -4 and y-intercept of -5, find an equation for the line and write your final answer in slope-intercept form. 5. Write an equation of the line that contains the point (-3,4) and is parallel to See example 7A on page 141 6. Write an equation of the line that contains the point (-2,4) and is perpendicular to See example 7B on page 141 Chapter 3, Section 1 1. Is every correspondence between two sets a function? Why or why not? No. A correspondence between two sets is a function only if exactly one element of the second set corresponds to each element of the first set. 2. What are the four different ways that we represented functions? 3. Can you explain what the domain and range of a function are? 4. What do the terms input and output refer to when working with functions? 5. How can we tell if an equation defines a function by looking at the graph of the equation? Practice Problems: 1. Look in your book at page 171-172 numbers 7-24. Can you identify which ones are functions and which are not. Also, which are one-to-one? Why are they one-to-one?
2. Page 173 numbers 27-34. 3. Page 173 numbers 47-62. 4. Page 174 numbers 75-80 Chapter 3, Section 2 1. How to find the domain and range from a graph of a function? See Example 2 on page 177 2. What does it mean for a function to be increasing or decreasing or constant? See the definition box on page 178 3. What does it mean for a function to be defined piecewise? When a function is defined by different expression. Practice Problems: 1. Page 184 number 9-20 2. Page 185 number 33-34 3. Page 185 number 37-36 4. Page 185 number 47-58 5. Page 186 number 59-64 Chapter 3, Section 3 Should Know: 1. It would be important to know and understand the table on page 195 labeled Graph Transformation Summary. 2. It would be important to know the six elementary graphs found on page 188-189. 3. How to tell if a function is even or odd or neither. See Example 6 on page 197 Practice problems: 1. Page 199 11-26 2. Page 199 numbers 27-36 3. Page 200 numbers 39-44 4. Page 200 numbers 45-78 Chapter 3, Section 4
Do you Know: 1. What is standard form for a quadratic function? 2. What is the vertex form of a quadratic function? 3. How to tell if a quadratic function is open up or open down? When does it have a max or min value? Open up if Open down 4. Using transformations, explain why the vertex of is (h,k). 5. Explain how to find the maximum or minimum value of a quadratic function. 6. What information does the constant a provide about the graph of the function in the form? Open up or down Practice problems: 1. Page 217 numbers 7-12 2. Page 218 numbers 25-34 3. Page 218 numbers 47-60 4. Page 219 numbers 69-74 Chapter 3, Section 5 1. Explain how to find the sum of two functions. 2. Explain how to find the product of two functions. 3. Is the domain of always the same as the intersection of the domains of and? Explain. Practice problems: 1. Page 232 numbers 29-42
2. Page 232 numbers 43-60 Chapter 3, Section 6 1. When a function is defined by ordered pairs, how can you tell if it is one-to-one? 2. When a function is defined by a graph, how can you tell if it is one-to-one? 3. What is the relationship between the graphs of two functions that are inverses? They are compositions of each other. That is Practice problems: 1. Page 247 number 13-24 2. Page 248 number 31-34 3. Page 249 number 45-54.