MATH 1130 Exam 1 Review Sheet The Cartesian Coordinate Plane The Cartesian Coordinate Plane is a visual representation of the collection of all ordered pairs (x, y) where x and y are real numbers. This collection of pairs is visualized by traveling to the first real number x along a horizontal number line (oriented left to right), then traveling to the second real number y along a vertical number line (oriented bottom to top). The two number lines, called the x-axis and y-axis, respectively, intersect at (0, 0), the point known as the origin. Distance Formula: Using the Pythagorean Theorem, we derived that the distance between two points (x 1, y 1 ) and (x 2, y 2 ) in the plane (meaning the length of the line segment connecting the two points) is (x1 x 2 ) 2 + (y 1 y 2 ) 2. Midpoint Formula: The coordinates of the midpoint of the line segment connecting two points (x 1, y 1 ) ands x 2, y 2 in the plane are given by ( x1 + x 2 2, y 1 + y 2 2 My advice with these formulas, as it is with almost every formula, is to not simply memorize them, because it is way too easy to just mix up the order of something, or mix up a + with a, forget an exponent, etc. Instead, think about what they mean and where they come from. The distance formula comes STRAIGHT from the Pythagorean Theorem. If it helps, actually draw the points on the plane and make a right triangle. The line segment connecting the two points will be the hypotenuse. For the midpoint formula, the x-coordinate of the midpoint is half way between the original two x-coordinates, and the y-coordinate of the midpoint is halfway between the two y-coordinates. I always say to myself average of the x s, average of the y s. ). References for Problems: Section P.6, Quiz 2 Problem 3 Graphs of Equations The graph of an equation in two variables, x and y, is the collection of all ordered pairs (x, y) in the Cartesian plane that make the equation true. For example, if the equation were x 2 + y 2 = 25, then the points (5, 0), (0, 5), and (3, 4) would be on the graph, but the point (1, 4) would not be. An x-intercept of a graph is a point on the graph that lies on the x-axis, which means y = 0. Analogously, a y-intercept of a graph is a point on the graph that lies on the y-axis, which 1
means x = 0. Given a graph, you can determine intercepts just by looking for them, but given an equation, you can find intercepts algebraically by setting the two variables equal to 0 individually. More generally, you can generate points on a graph by choosing a specific value for one of the variables, and then solving for the other. For example, for the equation 2x = 4y 2 6, you could set y = 0 and get x = 3, so ( 3, 0) is on the graph. If you set y = 1 you get x = 1, so ( 1, 1) is on the graph, and so on. References for Problems: Section 1.1, Quiz 2 Problem 4 Solving Linear Equations A linear equation in one variable x is an equation that can be written in the form ax+b = 0, where a and b are real numbers. We discussed that if a and b both turn out to be 0, then the original equation is always true (known as an identity), whereas if a turns out to be 0 and b does not, the original equation is NEVER true (known as a contradiction). I will not emphasize that terminology on the exam, rather we will focus on when a does not turn out to be 0, because those equations always have exactly one solution, which you should be able to find algebraically. References for Problems: Section 1.2, Quiz 2 Problem 5 Modeling with Linear Equations We did many examples in class of taking a real-life question, described verbally, and translating it into a equation. By plugging in all of the given information, and giving a variable name to the quantity you wish to solve for, you can then carry out algebra and find the answer. To see this in action, you can refer from your notes from that day, as well as look carefully through Section 1.3 in your text. We spent some time focusing on one particular kind of word problem, which we called mixing problems and required us to make one additional observation about the relationship between variables. For example, if a person sells 50 total cars, some of which are Corvettes and the rest of which are Malibus, then if C is the number of Corvettes and M is the number of Malibus, then C + M = 50 hence M = 50 C. References for Problems: Section 1.3, Quiz 3 Problems 1 and 2 Quadratic Equations A quadratic equation in a single variable x is an equation that can be written in the form ax 2 + bx + c = 0, where a, b, and c are real numbers and a 0. 2
There are a few techniques for solving quadratic equations: Factoring: Once all terms are moved to one side of the equation, and the other side is 0, you can (if possible) factor the quadratic, and solve the equation by setting the individual factors equal to 0. For example: x 2 5 = 4x x 2 4x + 5 = 0 (x 5)(x + 1) = 0 x 5 = 0 or x + 1 = 0 x = 5 or x = 1 Extracting Roots: If all of the dependence on the variable x lies inside of a square, then you can use the fact that u 2 = a means u = ± a to eliminate that square. For example: (x + 6) 2 5 = 0 (x + 6) 2 = 5 x + 6 = ± 5 x = 6 ± 5 Completing the Square: If the quadratic equation does not look like the one in the previous example, you can MAKE it look like that, by adding the appropriate number to both sides of the equation. Specifically, you can complete the expression x 2 + bx to (x + (b/2)) 2 by adding (b/2) 2 to both sides of the equation. For example: x 2 8x + 13 = 0 x 2 8x = 13 x 2 8x + 16 = 13 + 16 (x 4) 2 = 3 x 4 = ± 3 x = 4 ± 3 Quadratic Formula: A method that can solve any quadratic equation, or determine that no real solutions exist, is the quadratic formula, which says that the solutions to ax 2 + bx + c = 0 (a 0), if there are any, are given by x = b ± b 2 4ac, 2a which we derived by abstractly completing the square. In particular, this shows that if b 2 4ac > 0, there are two real solutions, if b 2 4ac = 0, there is one real solution, and if b 2 4ac < 0, there are no real solutions. References for Problems: Section 1.4, Quiz 3 Problems 3 and 4 3
Non-linear, non-quadratic equations that we can still solve We saw several examples of equations that did not initially have the structure of a linear or quadratic equation, but that we can still solve using the tools we have developed. Some types are: Certain higher degree polynomial equations: Some higher degree polynomial equations can be quickly simplified by factoring out common powers of x. Others can be factored as if they were quadratics, but with something other than x (for example x 2 ) playing the usual role of the variable. Rational Equations: Equations involving rational expressions (fractions) can be simplified by clearing denominators, meaning multiplying both sides of the equation by a common denominator. This still will eliminate all fractions, and what is left may be a linear or quadratic equation that you can solve. Be careful though, make sure the solutions you find would not result in division by 0 at an earlier stage. Equations with Radicals and Rational Exponents: When encountering an equation with square roots, try squaring both sides to eliminate the square roots. Similarly, when encountering an equation with a rational (fraction) exponent, rewrite the expression in terms of powers and roots (for example x 2/3 = 3 x 2 ), then raise both sides of the equation to the appropriate power to eliminate the roots. References for Problems: Section 1.6, Quiz 4 Problem 1 Linear Inequalities in One Variable To solve a linear inequality in a single variable, you proceed in the same way that you would to solve a linear equation, with one key caveat: If you multiply or divide both sides of an inequality by a negative number, the inequality direction changes. As long as you keep this in mind, solving inequalities is the same as solving equations, and you can even do two at once, for example: 5 < 1 4x 17 4 < 4x 16 1 > x 4 You should also be prepared to provide answers in interval notation, so in this case that would be [ 4, 1). References for Problems: Section 1.7, Quiz 4 Problem 2 4
Linear Equations in Two Variables Every non-vertical line in the Cartesian Plane is defined by a unique equation of the form y = mx + b, where m and b are real numbers. We call this form the slope-intercept form of the line, because we see from plugging in x = 0 that the line has its y-intercept at (0, b). Further, the number m is called the slope because it measures the steepness and direction of the line: the larger the absolute value of the slope, the steeper the line, and while a line with positive slope heads uphill from left to right, a line with negative slope heads downhill from left to right. Given any two points (x 1, y 1 ) and (x 2, y 2 ) on a non-vertical line, the slope can be determined with the formula m = y 2 y 1 x 2 x 1. This formula is often referred to more intuitively as change in y over change in x or rise over run. A way I like to interpret slope is that it is the change in elevation caused by moving along the line one unit to the right. While every non-vertical line has one slope-intercept form, it has many more perfectly good equations that define it. For example, if a line has slope m and passes through the point (x 1, y 1 ), then it is defined by the equation which we refer to as point-slope form. y y 1 = m(x x 1 ), If two lines have slopes m 1 and m 2, respectively, then we say that lines are parallel if m 1 = m 2, and perpendicular if m 1 m 2 = 1. References for Problems: Section 2.1, Quiz 4 Problems 3, 4, and 5 Functions and Their Graphs A function from a set A to a set B is a rule that assigns exactly one element of B to each element of A. More intuitively, a function is an input-output machine: you plug in something in A, it spits out something in B. The collection of everything you are allowed to plug in to the function is called the domain, and the collection of everything the function spits out is called the range. Remember our birthday example: if the domain is the people in our class, and the function is the person s birthday, then the range is NOT the collection of all days of the year, it is the collection of days that are actually the birthday of someone in our class. 5
In this class we focus on functions whose inputs and outputs are real numbers. In class, we saw several ways of describing such a function, and we will focus on the following: Input/Output Table: The weakness of defining a function using a table is that you can only define it for finitely many inputs. Example: input output 2 7-1 1.5 0 7 To check if a table really does define a function, make sure there is not an instance of the same input appearing more than once with different outputs. If such an occurrence can be found, the table does NOT define a function. As long as that does not happen, we have a function, and the domain is simply all the numbers that appear as inputs, in this case {2, 1, 0}, the range is all the numbers that appear as outputs, in this case {7, 1.5}, and the graph is just the points corresponding to the given pairs, in this case just the three points (2, 7), ( 1, 1.5), (0, 7). Function Notation: Example f(x) = x. This formula tells you explicitly how to x 2 1 compute the output for each input x, such as f(5) = 5/24, f( 2) = 2/3. If the domain is not specifically provided, then it is implied that the domain is all of the inputs x for which the formula is defined, which in this case only forbids x 2 1 = 0, in other words x = ±1, so the implied domain is all real numbers OTHER than 1 and 1. The nice thing about this kind of function definition is that you do not have to check if it defines a function or not, because it tells you exactly how to fine the ONE output for every permissible input. Equation in Two Variables: Example y 3 x 2 = 11 defines y as a function of x. For each input x, you can plug it in and solve for the output y. To check if an equation in two variables defines a function, try to solve for y by itself algebraically, and verify that you only get one value. In this example, we could solve for y and get y = 3 11 + x 2, so we do have y as a function of x, whereas if we made a minor change to the example and did y 4 x 2 = 11, then solving for y yields y = ± 4 11 + x 2, so we do not have y as a function of x because each input yields two outputs. Graphically: Remember we graph functions by plotting the input on the x-axis, and then plotting the output on the y-axis. Since each input can only have one output, we can tell if a given graph defines y as a function of x using the vertical line test, which says that a graph defines y as a function of x if and only if the graph intersects every vertical line at most once. Put another way, if a graph crosses any vertical line MORE than once, then it fails the vertical line test and we do not have y as a function of x. 6
If a graph passes the vertical line test, then to determine the domain of the function we ask: which x-values are being used by this graph? I like to visualize this by imagining the entire graph is flattened to the x-axis, and determining what points on the x-axis would be covered. Analogously, the range is the collection of y-values used by the graph, so we can imagine the graph flattened to the y-axis and determine what points on the y-axis would be covered. The zeros of a function f are all the inputs x such that f(x) = 0. If given a formula, you can set the output equal to 0 and solve for the inputs algebraically. If given a graph, you can just look for points on the x-axis, in other words x-intercepts, since those are the points at which y = 0. We say that a function f is increasing on an interval I if f(b) > f(a) whenever b > a and a and b are in I. Intuitively, this means that the graph is going uphill from left to right over the entire interval I. Analogously, f is decreasing on I if f(b) < f(a) whenever b > a and a and b are in I, meaning the graph is going downhill over the entire interval I. You should be prepared to, given a graph, determine the intervals on which a function is increasing and decreasing. Difference Quotients and Average Rate of Change: If a function f is defined on a closed interval [a, b], then the average rate of change of f on [a, b] is f(b) f(a), b a which is the slope of the line segment connecting the two points (a, f(a)) and (b, f(b)) on the graph. We did examples where this was computed with specific numbers a and b, and then we also did some algebraic manipulations of expressions for this slope in terms of variables. We call the latter expressions difference quotients, which take forms such as f(x + h) f(x) h or h(x) h(5). x 5 In these cases, the goal is to simplify the expression algebraically enough that the original denominator (in theses examples h and x 5, respectively) can be canceled. References for Problems: Sections 2.2 and 2.3, Quiz 5 7