NOTES [Type the document subtitle] Math 010
Cartesian Coordinate System We use a rectangular coordinate system to help us map out relations. The coordinate grid has a horizontal axis and a vertical axis. Where these two axes intersect is called the origin. The grid is also divided into 4 quadrants. Traditionally, we use Roman numerals to label these 4 quadrants. Let s begin by plotting some ordered pairs. A=(1, 4) C=(-,5) E=(-,-4) G=(,-1) B=(0,) D=(-4,0) F=(0,-) Find the ordered pair associated with the given points.
We also use the coordinate system to graph solutions of equations in two variables. One of the most common equations we graph is linear equations. LINES There are several methods to graph lines: plot points by creating a table of values, plot the intercepts, use the slope and y-intercept. How do you know if the graph of an equation is a line? y x 5 y 4x 11 y x 1 x 9 0 y 5 y x 5 Make a Table of Values to graph the following: x y 6 5xy 0 What happens when we do NOT have a linear equation? What happens when x is squared and y is not?
Make a Table of Values to graph the following: yx y x What happens when y is squared and x is not? Make a Table of Values to graph the following: x y x1 y
What happens when x is under a square root sign? Make a Table of Values to graph the following: y x y x
Relations and Functions A relation is a set of ordered pairs. The domain is the set of all first coordinates and the range is the set of all second coordinates. Bob Carly Juan Bow Tie Flower Scarf {(Bob, tie), (Carly, bow),(juan, flower),(bob, scarf)} The domain would be {Bob, Carly, Juan} The range would be {tie, bow, flower, scarf} A function is a relation such that no two ordered pairs have the same first coordinate. An example of a function is: {(Sally, pink), (Carly, blue), (Adam, green), (Bob, blue)} The domain would be {Sally, Carly, Adam, Bob} The range would be {pink, blue, green} Another example of a function is: A {( x, y) : y x 5, x,0,} B ( x, y) : y x 10, x 6, 1,6 Find the domain and range.
What if we have an infinite number of ordered pairs? We cannot make a list, but we can draw a picture of the relation. What is the domain? What is the range? How can we determine if a graph is a function? Remember definition: no two ordered pairs have the same first coordinate. This leads to the vertical line test. Do the following graphs represent a function? Find the domain and range.
In algebra, we use function notation: Function Notation Non-function notation: y x Re-written with function notation: f( x) x We read this as f of x f() () f() 9 f () f ( ) f ( ) Let f( x) x 1. Find f ( ), f (0), fa, ( ) f( a h ), f( a h) f( a) h Let g( x) x. Find g ( ), g (0), ga, ( ) g( a h ), g( a h) g( a) h
Linear Functions-Part 1 Part1: Solve for y. Identify the slope and y-intercept. The slope-intercept formula of a line is: y mx b Where m is the slope of the line and b is the y-intercept. Solve for y. Identify the slope and y-intercept. Ex #1. x4y 10 Ex #. x 1 y 5 Ex #. 5y 15 0
Linear Functions Part Part : Find the x-intercept and y-intercept. The x-intercept is where the graph crosses the x-axis and the y-intercept is where the graph crosses the y-axis. Find the x-intercept and y-intercept. Ex. #1: 4xy 8 Ex. #: x5y 0 Ex. #: y7x 1 Ex. #4: x y 1 4 5
Linear Functions Part Part : Find the slopes of the lines passing through the following points. y y1 Formula for slope: m x x 1 m rise run Find the slopes of the lines passing through the following points. Ex #1: (7,0) and (0,4) Ex #: (, 5) and (1,9) Ex #: (, 5) and ( 1, 5) Ex #4: (7, ) and (7,5)
Linear Functions Part 4 Part 4: Graphing lines using the slope and y-intercept. 1. Solve the equation for y.. Identify m and b.. Plot b on the y-axis. rise 4. From b, use the slope to get more points. run Ex. #1: 4x y 1 Ex. #: xy 9 Ex. #: 7 yx Ex. #4: y Ex. #5: x 1 0
Equations of Lines-Part 1 1. Find the equation of a line given the slope and y-intercept.. Find the equation of a line given the slope and a point. Find the equation of the line with the given information. Write answers in slopeintercept form, if possible. You will need to know formulas: Ex. #1: 1. Slope-intercept formula: y mx b. Point-Slope formula. y y m( x x ) 1 1 m ; y-intercept = 5 Ex. #: m 0; y-intercept = 1 5 Find an equation of a line given a slope and a point: Use the Point-Slope Formula: y y1 m( x x1) m slope Point ( x1, y 1) Ex. #: m 5; through (,1) Ex. #4: m ; through ( 4, ) 5
Extra Practice: m ; through (4, 1) Horizontal Equation: y number m 0 only has a y-intercept Vertical Equation: x number m is undefined only has an x-intercept Ex. #5: m 0; through ( 5,) Ex. #6: m is undefined; through (, 7)
Equations of Lines Part Find the equation of the line passing through the given points. x x1 1. Find the slope first. m y y 1. Pick one point and now use the Point-Slope formula. y y1 m( x x1). Write answers in slope-intercept form, if possible. Ex. #1: Passing through the points ( 1,) and (4,7) Ex. #: Passing through the points (, 4) and ( 5, 1) Ex. #: Passing through the points (5, 6) and (, 6) Ex. #4: Passing through the points ( 7, 4) and ( 7,8)
Parallel Lines Parallel Lines have the same slope. Find the equations of the lines passing through the given points parallel to the given line. Write answers in slope-intercept form when possible. Ex #1: Through (,5); parallel to x7y 14 1. Find the slope of the given line by solving for y.. Use the slope and the given point to write equation of line.. Write answers in slopeintercept form when possible. Ex #: Through ( 4, 9) ; parallel to y Ex #: Through (7, ) ; parallel to x 8
Perpendicular Lines Perpendicular Lines slopes are opposite reciprocals. (flip and change the sign) Find the equations of the lines passing through the given points perpendicular to the given line. Write answers in slope-intercept form when possible. Ex #1: Through (,5); perpendicular to y4x 5 Ex #: Through ( 7,) ; perpendicular to x5y 15 1. Find the slope of the given line by solving for y.. Find the opposite reciprocal of the slope. We label this m. Use m and the given point to write equation of line. 4. Write answers in slope-intercept form when possible. Ex #: Through ( 4, 9) ; perpendicular to y 8 Ex #4: Through (7, ) ; perpendicular to x
Linear Inequalities Graph the solution set of the linear inequalities: Steps: 1. Solve for y. Identify the slope and y-intercept.. Graph the line by plotting the y-intercept first (on the y-axis) and then use the slope to rise get other points, run.. Use a solid line if you have or. Use a dashed or dotted line if you have or.. Look at the y-intercept. Shade below the y-intercept if less than. Shade above the y-intercept if greater than. Ex. #1: yx Ex. #: 9 4x y Ex. #: 4x1 y Ex. #4: x Ex. #5: y 4
Linear Systems in Two Variables SUBSTITUTION METHOD: 1. Choose one of the equations and solve for one of the variables. Try to pick a variable with a coefficient of 1 or 1. This will eliminate having a lot of fractions.. Substitute this expression into the OTHER equation.. Solve the resulting equation. (It should now have only one variable) 4. Take this solution and substitute it into the expression obtained in step one. 5. Write your answer as an ordered pair. Ex. #1: Solve by the substitution method: 4x y4 x y Ex. #: Solve by the substitution method: x y 9 4x y 14
ELIMINATION METHOD: 1. If necessary, rewrite both equations in the form of Ax + By = C.. Multiply either equation or both equations by appropriate numbers so that the coefficients of x or y will be opposites with a sum of 0.. Add the equations. 4. Solve this equation. 5. Substitute this solution back into one of the ORIGINAL equations. 6. Write your answer as an ordered pair. x 7y 1 Ex. #: Solve by the elimination method: x y Ex. #4: Solve by the elimination method: 4xy5 x 8y10 SPECIAL SOLUTIONS: If both variables are eliminated when you are solving a system then 1. There is NO SOLUTION when the resulting statement if false.. There are INFINITELY MANY SOLUTIONS when the resulting statement is true. (They were the same line) Ex. #5: Solve by the substitution xy5 method: x 4y 7 Ex. #6: Solve by the elimination x6y14 method: 5x15y5
Linear Systems in Three Variables SOLVING A SYSTEM FOR THREE UNKNOWNS 1. Pick two equations and eliminate ONE variable. Put a star by this result.. Pick another two equations and eliminate the SAME VARIABLE. Put a star by this result.. Put these two resulting equations (the star equations) together and solve these by the addition method. 4. Substitute this solution back into one of the star equations. 5. Now substitute both of these solutions back into one of the ORIGINAL equations. 6. Write your answer as an ordered triplet. x y 4z 6 7x y z x y z 7
Applications of Systems Part 1 Solve word problems using a system of equations. Ex. #1: The sum of two numbers is 6. The larger number is 5 more than twice the smaller number. What are the numbers? 1. Define the variables.. Write the equations.. Solve. 4. Answer the problem. Ex. #: The difference of two numbers is 1. Twice the larger number plus the smaller number is 18. What are the numbers? Money problems: If I have quarters, I have 75. (value of coin) = $money If I have 9 nickels, I have 45. 9(value of coin) = $money (How many of the item)(value of the item)= $ Total money Ex. #: Jan has $5.90 in dimes and quarters. She has a total of coins. How many dimes and how many quarters does Jan have? Ex. #4: Elaine spent $5.96 on 1 and 6 stamps. She bought a total of 1 stamps. How many 1 stamps and how many 6 did she buy?
Applications of Systems Part Solve word problems using a system of equations. Ex. #1: Alex sold tickets to the high school football game. Adult tickets cost $4.00 and student tickets cost $1.50. Alex collected $10 from the sale of the tickets. How many adult tickets and how many student tickets did Alex sell? 1. Define the variables.. Write the equations.. Solve. 4. Answer the problem. Ex. #: Yolanda bought pens and 1 notebook for $1.70. Four days later she bought pens and notebooks for $.6. What is the cost of each pen and each notebook?
Greatest Common Factor GCF: The largest number that will divide into all the terms AND the variable(s) that occur in all the terms (raised to the smallest power). Find the GCF of the following polynomials: 4 1. 14x 1x 8x. 16a 4b 8a Factor the following:. 8x 18xy x 4. 7x y 5x y 14x y 4 5 5. 7 y(x 9) 14 z(x 9) 6. 15 x( y ) 5 z( y)
Factor by Grouping GROUPING: This factoring technique is used when you have 4 or more terms. Try grouping the first two terms and the last terms. Factor out the GCF of each group. If the GCF matches exactly, then take out that GCF from both groups. Factor further, if necessary. Factor the following: 1. 10hn km hm 15kn. 4x z 6x z xz 14xz 4. 7 0 70 xy xy y x y
ALWAYS look for a GCF first. Difference of Squares: 1. Factoring: Difference of Squares a b ( a b)( a b) 5x 81y. 4 16x 1. x 50 4. ( x ) 9 5. (x y) 49z 6. x (5x 4) 9(5x 4)
Cubes: 1, 8, 7, 64, 15, 16, 1000 Factoring: Cubes Sum of cubes: a b ( a b)( a ab b ) Difference of cubes: Sum of cubes: a b ( a b)( a ab b ) x 8 Difference of cubes: x 15 ALWAYS look for a GCF first. 1. 15x 7y. 40x 15y. 4 y y y 64 18 4. 5 1x y 75x y 1x y 75xy
TRINOMIAL: A polynomial with terms. Factoring Trinomials ALWAYS look for a GCF first. It is a good idea to make leading coefficient positive. Guess and check method is one of many methods, but not the only one. Factor the following: 1. x 14x 45. x 6x 10. 8x 16x 40 4. 6x 7x 0 5. x 5x 1 6. 4 x x 7. 8x 6xy 5y 8. x 4x 5 4
Solve Quadratic Equations by Factoring A quadratic equation has the form of ax bx c 0 Steps to solve by factoring: 1. Write in standard form: ax bx c 0. Factor.. Set each factor that has a variable equal to zero. 4. Solve each resulting linear equation. 1. 14x 1x 0. 1x 6x10 0. 5x 8x 1 x( x 8) 4. 7x x 4 ( x 4)( x 4)
Evaluating Integer Exponents 5 Negative Exponents: Take the reciprocal of base and change the sign of the exponent. x y = x y = Zero Exponent: 0 5 = 0 x = 4 0 (10 xy ) = ANYTHING (except zero) raised to the power of zero equals 1. Evaluate each of the following. 1. 5. 5 4. ( ) 4 4. ( 5) compared to 5 5. 1 6. 1 1 7. 8. 1 0 5 5 5
Simplifying Integer Exponents Review Negative Exponents: Take the reciprocal of base and change the sign of the exponent. x y = x y = Zero Exponent: 0 x = 4 0 (10 xy ) = Simplify each of the following. Your answers should have no NEGATIVE exponents. 1. Multiply the exponents to get rid of parentheses.. Make all exponents positive.. Clean up. 1 1. x y 5xy 5 4 xy. x y. 5xy 8xy 5 6 1 4. 6xy 9 x y 0 5 6
Review Reducing Fractions: Reducing Rational Expressions 5 0 7 9 x x 5 5 1. Factor.. Cancel common FACTORS. 1. 8xy z 5 4 x y z. x y 14 xy x y xy. x 6x16 64 x 4. 8x 7 x x9
Multiplying and Dividing Rational Expressions Directions: Perform the indicated operations and reduce to lowest terms. Even though the "indicated operations" are multiplication and division, what we need to do is FACTOR and REDUCE. 1. 6 7 10m p 5 4x y 18xy 9x y m p. 4 x 5x0 6x 9 xz z 1. Factor. Reduce. x x 15 x x 10 x 9x 10 1x 7x
Adding and Subtracting Rational Expressions Part 1 When adding or subtracting fractions that have the same denominator: 1. Keep the denominator and collect like terms in the numerators.. Try to reduce the expression by factoring. Directions: Perform the indicated operations and reduce to lowest terms. 1. 5 4. x 1 4 x 6 x x 5x 5x. x x 6 x x 4. 5 6 x x OPPOSITE DENOMINATORS x 5x 5. y y 6. x 7 x x 1 x4 4x
Adding and Subtracting Rational Expressions Part When adding or subtracting fractions that have different denominators, one must find the least common denominator (LCD) before adding or subtracting the fractions. Process to find the LCD: 1. Factor each denominator.. Write down one of every kind of factor.. Raise each factor to its highest power. Find the LCD: 5 7 a b c ab c d 5 7 6x y 4x y 4 5 5 Perform the indicated operations and reduce to lowest terms: 5 1 1. x 8x. 6 x1 x4 1. Find the LCD. Re-write each fraction with the LCD. Collect like terms of numerators 4. Reduce, if possible.. x 1 x 4 x x4 x7 x1 x x 4 x x 0 4. x x6 x 9x 10 x 6x 8 5.
Synthetic Division Synthetic Division is a condensed method of long division. It is quick and easy. Unfortunately, it can only be used when the divisor is in the form of ( x a) Review long division: 9x 5x1 x 1 Synthetic division: 9x 5x1 x 1 x 5x x 1 4 x 5x 10x x x 15 x 5 Reminders: 1. Write both polynomials in standard form.. Fill in all missing terms with a place holder of zero.. Write your answer as a polynomial that is one degree less than the dividend (numerator).
Complex Fractions A complex fraction is a fraction that has fractions in the numerator and/or the denominator. Single Fractions: Directions: 1. 1x 8y 7x 16y 4 Simplify the following fraction.. x 5x6 10x 5 4 x 6x Change the division to multiplication. Reduce.. 4. 1 x y 6 5xy 1 1 x 5 5 7x 14xy Multiple Fractions: 1. Find the LCD of all the fractions.. Multiply every term by the LCD.. Reduce
5. 7 10 x x 8 5 4 x x Multiple Fractions: 1. Find the LCD of all the fractions.. Multiply every term by the LCD.. Reduce 6. 1 x y x 7. 5 h 5 h h
Rational Equations 1. Find the L.C.D.. Multiply EVERY term by the LCD to get rid of all the fractions. (OR Cross-multiply, if you can**). Solve the resulting equation. 4. Check for extraneous solutions. (Substitute answers into the denominator to see if this would cause division by zero. You must throw out any solutions that cause division by zero because it is undefined.) 1. 7 5 1 6x 8x 1. 1 1 1 x 1x 1. 4 16 x 1 x x x 4. 6 x x x x
x 5 5. x 1 x x ** 6. 5 5x 5x x1 ** 7. x 4 x 1 x x
Applications Rational Equations Consecutive integers: Remember that integers are only negative and positive whole numbers. They do not include any decimals or fractions. Two consecutive integers: x, x+1 Two consecutive odd integers: x, x+ Two consecutive even integers: x, x + Reciprocals: If the number is x, then its reciprocal would be 1 x. Define the variable. Write an equation. Solve. 1. The sum of a number and its reciprocal is 9. What is the number? 10. The difference of the reciprocals of two consecutive integers is 1. What are the integers?
. The sum of the reciprocals of two consecutive even integers is 9. What are the 40 integers? Work: 1 1 1 time alone time alone time together 4. Paris can wash her car in 4½ hours. Her friend, Celia, can wash the same car in 7 hours. Working together, how long will it take them to wash the car? 5. Working together, Joseph and Dylan can write the computer program in 11 hours. Working alone, Joseph can write the computer program in 15 hours. How long does it take Dylan to write the program by himself?
Radicals What does a radical sign look like? Here are some examples:,, 4, 5 Square root: 49 1 81 16 Cube root: 8 Fourth root: 4 16 7 1 64 4 ( ) 4 4 16 Fifth root: 5 5 5 ( ) 5 Even root of a negative number is NOT real. Convert rational exponents to radicals: Odd root of a negative number is a negative number. 4 9 8 5 (5 x y ) Convert radicals to exponents. Simplify where possible. 49 8 5 4 7 5 x 5 0 p 1 (5 xy )
Rational Exponents Evaluate each of the following, if possible. 1. Make any negative exponents positive.. Change to radicals. Simplify. 1. 1 100. ( 5) 1. 5 1 4. 64 5 5. 81 49 6. 1 5 7. 1 5 8. ( 5) 1 9. ( ) 5 10. 1 1 6 8 8 11. 8 8 1 1 6 Simplify. All answers should have only POSITIVE exponents. 1. x 1 1 x 1. 5x 5 x 4 14. 1 5 4 5x 10x 1 4 y y
Simplify Radical Expressions Part 1 Simplify the following: 1 98 150 1. Prime factor the number.. For square root: Look for pairs. For cube root: Look for of a kind. For 4 th roots: Look for 4 of a kind. etc.. "Take out" the pairs, of kind, etc. 40 54 40 4 48 4 40 xy xy 4 4 5 6 x y z 4 100 5 1 x y z
Simplify Radical Expressions Part Rationalize the Denominator Rationalize the denominator means to eliminate any radicals in the denominator. A process to follow is: 1. Reduce the fraction, if possible.. Simplify the radicals. Rationalize by multiplying by "what you need". 4. Reduce again if necessary. Simplify the following: SQUARE ROOTS: 50 1... y 4. 5x 0x 1 5. 1x 6. 49x 9y CUBE ROOTS: 5 7. y 8. 5 9y y 9x 9. 5
Adding & Subtracting Radical Expressions Part 1 Review of collecting like terms: x 5x x 5 x Perform the indicated operations: 1. 4 1. Simplify all radicals. Add the coefficients of "like" radicals.. 40 90 160. x 18y x x y 4y 1x xy 4. 50xy 54xy
Adding & Subtracting Radical Expressions Part 1. Perform the indicated operations: 4 7 1. Simplify radicals.. Rationalize all denominators.. Find the LCD 4. Re-write all terms with LCD. 5. Combine like terms.. 5 6 4. 7 4. 5 6
Multiply Radicals Multiplying a monomial by a monomial: Multiply the "outsides" Multiply the "insides" Simplify, if possible. ( )(4 15) ( y)(5 x ) Multiply a square root by the SAME square root: ( )( ) ( 7 y)( 7 y ) ( 5 y)( 5 y) Perform the indicated operations and simplify your answers: 1. 5y 5y x 1. F.O.I.L.. Simplify. Combine like terms.. (4 )(5 8 6). x y 4. (5 x ) 5. ( 5x 1)
Divide Radicals Review: ( x5)( x 5) ( x 5)( x 5) ( 5)( 5) The "conjugate" of x 5 is x 5 The "conjugate" of 5 is 5 As you can see, the conjugate is found by changing the middle sign. When you multiply conjugates, you just need to square each term and then subtract. We use the conjugate to rationalize the binomial denominators. Rationalize the denominator of the following. 1 1.. 51 6. 8 6 8 4. x y x 5y
Radical Equations Process: 1. Isolate the radical.. Get rid of the radical by raising both sides to the appropriate power. x x. Solve the resulting equation. 4. Check for extraneous solutions. x x 4 4 x x 1. 4x 1 5 0. x 4 1 4. x 16 6 1 4. 4 x x 4 5. x 4x 15 6. x 4 x
Extra Example: 0 x x Remember that a fractional exponent can be written in radical form. 5 5 x 5 x or x x x or x If you encounter an equation that has a variable raised to a fractional exponent, you solve it just like a radical equation. Get rid of the radical by raising both sides to the appropriate power. x x 5 5 x x 7. x x 6 7 7 8. x 9
Complex Numbers What happens when we want to solve the equation: x 4? In order to solve this equation, we must introduce 1 and the set of imaginary numbers. We will represent 1 with i. This leads to 1 i 1 1 i Therefore, 1 i Any square root of a negative number can be written in terms of i. 4 9 1 A complex number has a real part and an imaginary part. 5i We can add, subtract, multiply, and divide complex numbers. Perform the indicated operations: 1. ( 4 7 i) ( i). ( 5 i) (7 4 i). ( i)(4 5 i) 4. (4 i) Replace i with 1
Rationalize the denominators. (Divide) Review: 5 5. 5 i i Review: 4 5 The "complex conjugate" of 5i is 5i Multiply: ( 5 i)( 5 i) 6. i 5i 7. i 54i
Quadratic Formula This is another method to solve quadratic equations. If the quadratic cannot be factored we have to have something else that will allow us to solve the equation. There are such methods completing the square and the quadratic formula. The quadratic formula is derived from completing the square on the general equation: ax bx c 0 You MUST memorize the formula: x a Process: 1. Write the equation in standard form:. Identify a, b, and c. b b 4ac ax bx c 0. Substitute numbers into formula. 4. Carefully do the arithmetic under the square root sign. 5. If possible, simplify the radical. 6. If possible, reduce the fraction. 1. x 4x1 0. 9x 18x 7 0. 9 11 4. x( x ) 6x 11 x x x 5. (x )( x 4) 7
Completing the Square This is another method to solve quadratic equations. If the quadratic cannot be factored we have to have something else that will allow us to solve the equation. There are such methods completing the square and the quadratic formula. Completing the Square is also used for other applications. Process: ax bx c 1. Write the equation in standard form:. Move c to the left hand side of the equation. 0 x bx c. If a is NOT = 1, divide all terms by a. Reduce any fractions. 4. Take 1 of the coefficient of x. 5. Square this and add to both sides of the equation. 6. Re-write left hand side as a squared binomial. 7. Solve the equation by the extraction of roots method. 1. x 8x11 0. x 6x18 0. x x1 0 4. x x 10 0 5. x 5x 7 0
Applications Quadratic Equations Consecutive integers: Remember that integers are only negative and positive whole numbers. They do not include any decimals or fractions. Two consecutive integers: x, x+1 Two consecutive odd integers: x, x+ Two consecutive even integers: x, x + Reciprocals: If the number is x, then its reciprocal would be 1 x. Define the variable. Write an equation. Solve. 1. Find two consecutive positive integers. Find two consecutive odd integers whose product is 1. whose product is 5.. The sum of the squares of two even consecutive integers is 40. Find the integers. 4. The sum of a number and its reciprocal is 6. What is the number? 5. Three times the square of a number is 6 more than twice the number. What is the number?
Area: Rectangle: Area = base x height or Area = length x width. Triangle: Area = 1 base x height or Area = base x height 6. The area of a rectangle is 5 sq. ft. The length is 1 ft. less than twice the width. What are the dimensions of the rectangle? 7. The height of a triangle is in. more than three times the base. Find the base and the height if the area of the triangle is 5 sq. in. Work: 1 1 1 time alone time alone time together 8. One pipe can fill a reservoir hours faster than another pipe can. Together they fill the reservoir in 5 hours. How long does it take each pipe to fill the reservoir?