COST PER UNIT Some problems will require the calculation of unit cost. Example If 100 square feet cost $1,000, how much does 1 square foot cost? = ᎏ $ 1 1 0 , 0 00 ft 0 2 ᎏ = $10 per square foot MOVEMENT In working with movement problems, it is important to use the following formula: (Rate)(Time) = Distance Example A courier traveling at 15 mph traveled from his base to a company in ᎏ 1 4 ᎏ of an hour less than it took when the courier traveled 12 mph. How far away was his drop off? First, write what is known and unknown. Unknown: time for courier traveling 12 mph = x. Known: time for courier traveling 15 mph = x – ᎏ 1 4 ᎏ . Then, use the formula (Rate)(Time) = Distance to find expressions for the distance traveled at each rate: 12 mph for x hours = a distance of 12x miles. 15 miles per hour for x – ᎏ 1 4 ᎏ hours = a distance of 15x – ᎏ 1 4 5 ᎏ miles. The distance traveled is the same, therefore, make the two expressions equal to each other: 12x =15x – 3.75 –15x = –15x ᎏ – – 3 3 x ᎏ = ᎏ –3 – . 3 75 ᎏ x = 1.25 Be careful, 1.25 is not the distance; it is the time. Now you must plug the time into the formula (Rate)(Time) = Distance. Either rate can be used. 12x = distance 12(1.25) = distance 15 miles = distance Total cost ᎏᎏ # of square feet – THEA MATH REVIEW– 130 WORK-OUTPUT Work-output problems are word problems that deal with the rate of work. The following formula can be used on these problems: (Rate of Work)(Time Worked) = Job or Part of Job Completed Example Danette can wash and wax 2 cars in 6 hours, and Judy can wash and wax the same two cars in 4 hours. If Danette and Judy work together, how long will it take to wash and wax one car? Since Danette can wash and wax 2 cars in 6 hours, her rate of work is ᎏ 6 2 h c o a u rs rs ᎏ , or one car every three hours. Judy’s rate of work is therefore, ᎏ 4 2 h c o a u rs rs ᎏ , or one car every two hours. In this problem, making a chart will help: Rate Time = Part of job completed Danette ᎏ 1 3 ᎏ x = ᎏ 1 3 ᎏ x Judy ᎏ 1 2 ᎏ x = ᎏ 1 2 ᎏ x Since they are both working on only one car, you can set the equation equal to one: Danette’s part + Judy’s part = 1 car: ᎏ 1 3 ᎏ x + ᎏ 1 2 ᎏ x = 1 Solve by using 6 as the LCD for 3 and 2 and clear the fractions by multiplying by the LCD: 6( ᎏ 1 3 ᎏ x) + 6( ᎏ 1 2 ᎏ x) = 6(1) 2x + 3x =6 ᎏ 5 5 x ᎏ = ᎏ 6 5 ᎏ x =1 ᎏ 1 5 ᎏ Thus, it will take Judy and Danette 1 ᎏ 1 5 ᎏ hours to wash and wax one car. Patterns and Functions The ability to detect patterns in numbers is a very important mathematical skill. Patterns exist everywhere in nature, business, and finance. When you are asked to find a pattern in a series of numbers, look to see if there is some common number you can add, subtract, multiply, or divide each number in the pattern by to give you the next number in the series. For example, in the sequence 5, 8, 11, 14 . . . you can add three to each number in the sequence to get the next number in the sequence. The next number in the sequence is 17. – THEA MATH REVIEW– 131 Example What is the next number in the sequence ᎏ 3 4 ᎏ ,3,12,48? Each number in the sequence can be multiplied by the number 4 to get the next number in the sequence: ᎏ 3 4 ᎏ ϫ 4 = 3, 3 ϫ 4 = 12, 12 ϫ 4 = 48, so the next number in the sequence is 48 ϫ 4 = 192. Sometimes it is not that simple. You may need to look for a combination of multiplying and adding, divid- ing and subtracting, or some combination of other operations. Example What is the next number in the sequence 0, 1, 2, 5, 26? Keep trying various operations until you find one that works. In this case, the correct procedure is to square the term and add 1: 0 2 + 1 = 1, 1 2 + 1 = 2, 2 2 + 1 = 5, 5 2 + 1 = 26, so the next number in the sequence is 26 2 + 1 = 677. P ROPERTIES OF FUNCTIONS A function is a relationship between two variables x and y where for each value of x, there is one and only one value of y. Functions can be represented in four ways: ■ a table or chart ■ an equation ■ a word problem ■ a graph – THEA MATH REVIEW– 132 For example, the following four representations are equivalent to the same function: Helpful hints for determining if a relation is a function: ■ If you can isolate y in terms of x using only one equation, it is a function. ■ If the equation contains y 2 , it will not be a function. ■ If you can draw a vertical line anywhere on a graph such that it touches the graph in more than one place, it is not a function. ■ If there is a value for x that has more than one y-value assigned to it, it is not a function. Word Problem Table Graph Equation Javier has one more than two times the number of books Susanna has. y = 2x + 1 x y –3 –5 –2 –3 –1 –1 0 1 1 3 2 5 1 4 3 2 –5 –1 –2 –3 –4 1 5 4 32 –5 –1 –2 –3–4 5 y x – THEA MATH REVIEW– 133 In this graph, there is at least one vertical line that can be drawn (the dotted line) that intersects the graph in more than one place. This is not a function. In this graph, there is no vertical line that can be drawn that intersects the graph in more than one place. This is a function. y 1 4 3 2 –5 –1 –2 –3 –4 1 5 4 32 –5 –1–2–3–4 x y 1 4 3 2 –5 –1 –2 –3 –4 1 5432 –5 –1 –2 –3–4 x 5 5 In this table, the x-value of 5 has two corresponding y-values, 2 and 4. Therefore, it is not a function. x y 5 2 3 –1 2 0 6 1 5 4 In this table, every x-value {–2, –1, 0, 1, 2, 3} has one corresponding y-value. This is a function. In this table, every x-value {–2, –1, 0, 1, 2, 3} has one corresponding y-value, even though that value is 3 in every case. This is a function. x y –2 5 –1 6 0 7 1 8 2 9 x y –2 3 –1 3 0 3 1 3 2 3 – THEA MATH REVIEW– 134 Examples x = 5 Contains no variable y, so you cannot isolate y. This is not a function. 2x + 3y = 5 Isolate y: 2x + 3y =5 –2x –2x ᎏ 3 3 y ᎏ = ᎏ –2x 3 +5 ᎏ y =– ᎏ 3 2 x ᎏ + ᎏ 5 3 ᎏ This is a linear function, of the form y = mx + b. x 2 + y 2 = 36 Contains y 2 , so it is not a function. |y| = 5 There is no way to isolate y with a single equation, therefore it is not a function. FUNCTION NOTATION Instead of using the variable y, often you will see the variable f(x). This is shorthand for “function of x” to auto- matically indicate that an equation is a function. This can be confusing; f(x) does not indicate two variables f and x multiplied together, it is a notation that means the single variable y. Although it may seem that f(x) is not an efficient shorthand (it has more characters than y), it is very elo- quent way to indicate that you are being given expressions to evaluate. For example, if you are given the equation f(x) = 5x – 2, and you are being asked to determine the value of the equation at x = 2, you need to write “evalu- ate the equation f(x) = 5x – 2 when x = 2.” This is very wordy. With function notation, you only need to write “determine f(2).” The x in f(x) is replaced with a 2, indicating that the value of x is 2. This means that f(2) = 5(2) – 2 = 10 – 2 = 8. All you need to do when given an equation f(x) and told to evaluate f(value), replace the value for every occur- rence of x in the equation. Example Given the equation f(x) = 2x 2 + 3x + 1, determine f(0) and f(–1). f(0) means replace the value 0 for every occurrence of x in the equation and evaluate. f(0) = 2(0) 2 + 3(0) + 1 = 0 + 0 + 1 =1 f(–1) means replace the value –1 for every occurrence of x in the equation and evaluate. f(0) = 2(–1) 2 + 3(–1) + 1 = 2(1) + –3 + 1 = 2 – 3 + 1 =0 – THEA MATH REVIEW– 135 . cost ᎏᎏ # of square feet – THEA MATH REVIEW– 130 WORK-OUTPUT Work-output problems are word problems that deal with the rate of work. The following formula can be used on these problems: (Rate of Work)(Time. sequence is 48 ϫ 4 = 192 . Sometimes it is not that simple. You may need to look for a combination of multiplying and adding, divid- ing and subtracting, or some combination of other operations. Example What. is 26 2 + 1 = 677. P ROPERTIES OF FUNCTIONS A function is a relationship between two variables x and y where for each value of x, there is one and only one value of y. Functions can be represented