Name
EVALUATING ALGEBRAIC EXPRESSIONS Objective: To evaluate an algebraic expression Example Evaluate the expression if and y = 5 6x y Original problem 6() ( 5) Substitute the values given into the expression and multiply + 0 Add Final answer Evaluate each expression. Show all work.. x ( y), when 5 and y =. 0 x ( y+ z), when, y = and z =. x + (5 y), when 5 and y =. 7 x ( x y), when and y = 6 5. x + y z, when, y =, and z = 6. x y + z, when, y = and z =
THE DISTRIBUTIVE PROPERTY Objective: To use the distributive property Example ( x + 5) + x Original problem ( x) + (5) + x Distribute the x 0+ x Simplify x 0 Combine like terms Simplify the following expressions. Show all work.. x (+ 5 x). ( 5) x xx. (x ) x. ( x 5) + ( x) 5. ( x+ 5) + x( x ) 6. 5 ( x ) + ( x )
PROPORTIONS Objective: To solve proportions Example a 5 a = Original problem (a 5) = a Cross-multiply 9a 5 = a Distribute 5a = 5 Combine like terms a = Final answer Solve each proportion for the indicated variable. Show all work.. x n n =. =. 0 x = x + x. x 7 x 8 = 5. 0 5 x + 6 = x x y 6. = y 6
SOLVING LINEAR EQUATIONS Objective: To solve a linear equation Example ( ) 5 x = x Original problem 6 x = x 5 5 Distribute the 5 5 6 x x = 5 5 Multiply the entire equation by 5 to clear the fractions x 6= 5x 0 Subtract 5x from both sides of the equation x 6= 0 Add 6 to both sides of the equation x = Divide both sides of the equation by Solve for x Solve each equation. Show all work.. 8 0 y =. 5 6. ( ) 5 x = x+. x = x 5+ x 5. ( y ) = ( y 5) 6. 5( x) = ( x ) = x + 8. 7. ( x) ( ) y = y + 9. x ( x ) = x
SOLVING LINEAR INEQUALITITES Objective: To solve and graph simple inequalities Example y 8< 0 Original problem y < 8 Add 8 to both sides y < 6 Divide both sides by Example Remember if you multiply or divide by a negative number when solving an inequality you must switch the way the inequality sign points. x 5x Original problem x 5x Add to both sides x Subtract 5x from both sides x Divide both sides by and reverse the inequality Solve and graph. Show all work.. x >. x 6x+. y< 7y. x 5. x > 6. x < + x 7. > y 8. x + 5 9. ( x ) x 5 5
FINDING SLOPE OF A LINE Objective : To find the slope of a line when given two points Objective : To find the slope when given an equation of a line Example Find the slope of the line containing (,) and (,) y y m = x x m = Substitute values into the formula ( ) m = + m = Simplify Simplify 5 Example Find the slope of the line x y = Solve the equation for y and put in slopeintercept form. y = mx+ b y = x+ Subtract x from each side y = x Multiply both sides by m = Slope is Find the slope of the segment defined by the given two points. Show all work.. (,) and (6, ). (,6) and (0,0). (,) and ( 5, ) Find the slope of each line. Show all work.. 9x+ y = 5. x+ y = 6. x+ 5y = 6
EQUATIONS OF LINES Objective: To write an equation of a line in standard form when given the slope and one point Example Write an equation of a line in standard form that passes through the point (, ) and has a slope of. y y = m( x ) Use the point-slope form to find the equation x y = ( x ) Substitute in for m, x, and y y = x Distributive Property y = x + Add to both sides Remember when adding fractions you must find a common denominator. Example: + 6 + y = x + Multiply by on both sides to clear the fractions x + y = Subtract x on both sides x y = Multiply equation by to put equation in standard form Find the equation of the line in standard form for the given slope and point. Show all work.. (,) m =. ( 0, ) m =. (, ) m =. (, ) m = 5. ( 6, 5) m = 6. (, ) m = 5 7
EQUATIONS OF LINES Objective: To write an equation of a line in standard form when given two points Example Write an equation of a line in standard form that passes through the points (,) and (5, ) y y m = Find the slope x x m = Substitute values into the formula 5 ( ) m = Simplify to find the slope(m) y y = m( x ) Use the point-slope form to find the equation x In this example the point (,) was used however the point (5, ) could also have been used. y = [ x ( )] Substitute one of the above points in for (, ) x y and for m y = ( x + ) Simplify Remember when adding fractions you must find a common denominator. y = x Distributive Property y = x+ Add to both sides Example: + + = y = x + Multiply by on both sides to clear the fractions x + y = Add x on both sides to put equation in standard form Find the equation of the line in standard form for the given two points. Show all work.. (6,) and (,6). (, ) and (5,5). (6,) and (0, ) 8
IDENTIFYING PARALLEL AND PERPENDICULAR LINES Objective: To determine if two lines are parallel or perpendicular Example Two lines are parallel if their slopes are the same but they have different y-intercepts. Example Two lines are perpendicular if their slopes are negative reciprocals of each other. Line contains (,5) and (,) Line contains (,) and (,) The slope of Line is The slope of Line is 5 = ( ) = ( ) same slope Line contains (,) and (,7) Line contains (,) and (,0) The slope of Line is 7 = 0 The slope of Line is = ( ) Negative reciprocal slope Determine if the lines are parallel, perpendicular, or neither. Show all work.. Line contains (,7) and (, 5) Line contains ( 6, 0) and (0, ). Line contains (, ) and ( 6,) Line contains (,5) and (5,). Line contains (, ) and (,) Line contains (6,) and ( 8,9) Write each equation in slope-intercept form and find the slope of each line. Compare the slopes and determine if the lines are parallel, perpendicular, or neither. Show all work.. x+ y = x y = 6 5. y = x x y = 6. x+ y = x y = 9
GRAPHS OF LINEAR EQUATIONS Objective: To use x and y-intercepts to sketch a quick graph of a line Example Use x and y-intercepts to sketch the line of x+ 5y = 5. To sketch the line, plot the intercepts, and draw a line through them. x-intercept x+ 5y = 5 Original problem x + 5(0) = 5 Find x-intercept by setting y = 0 5 x-intercept is (5,0) y-intercept x+ 5y = 5 Original problem (0) + 5y = 5 Find y-intercept by letting x= 0 y = y-intercept is (0,) Sketch the graph of each line using the x and y-intercepts. Show all work.. 9x+ y = 8. 5x 0y = 0. 6x+ y = 8 x-int: x-int: x-int: y-int: y-int: y-int: 0
GRAPHS OF LINEAR EQUATIONS Objective: To use slope-intercept form of a line to sketch a quick graph of a line Example Sketch the graph of the line x+ y =. Begin by solving the equation for y x+ y = Original problem To sketch the line, first plot the y-intercept (0, ) then locate the second point by moving unit down and units to the right. Finally, draw the line through the two points. y-intercept y = x+ Subtract x from both sides y = x+ Divide both sides by m=, b= List the slope(m) and the y-intercept(b) Sketch the graph of each line using the slope and the y-intercept. Show all work.. x y =. x + y = 9. x + y = m = m = m = y-int: y-int: y-int:
SYSTEMS OF EQUATIONS (SUBSTITUTION) Objective: To solve a system of equations using substitution Example x y = 9 x + 5y = x y = 9 y = x + 9 y = x 9 x + 5y = x + 5(x 9) = x+ 5x 5= 7x 5 = 7x = Original system of equations Solve one of the equations in terms of a variable Substitute x 9 in for y in the other equation Solve for x Hint: Pick the equation that has a variable with a coefficient of. x y = 9 () y = 9 6 y = 9 y = Substitute the value for x in to the other given equation to find y Solve for y (, ) Write your answer as an ordered pair
Use the substitution method to solve the system of equations. Show all work.. x y = 5x 7y =. x+ y = x+ y =. 0x 6y = 7 x+ y =. x y = 6 x+ y = 9 5. x+ y = 7 x y = 7 6. x y = x+ y = 5
SYSTEMS OF EQUATIONS (LINEAR COMBINATION) Objective: To solve a system of equations using linear combination Example x + 7y = x y = x+ 7y = ( x y = ) x + 7y = x y = y = y = Original system of equations Multiply the second equation by so that x will cancel out Add the two equations together Solve for y x+ 7y = To find x substitute the value of y (found in the last step) into one of the x + 7() = original equations x + 7= Simplify 8 Solve for x (,) Write your answer as an ordered pair Example x+ 5y = 6 x+ y = 5 Original system of equations If no variable in either equation has a coefficient of then use this method. (x+ 5y = 6) Multiply the first equation by ( x+ y = 5) Multiply the second equation by x + 0y = x+ 6y = 5 6y = 9 y = Solve for y Add the two equations together x+ y = 5 To find x substitute the value of y into one of the original equations x + = 5 x + = 5 Simplify x = Solve for x, Write your answer as an ordered pair
Use the linear combination method to solve the system of equations. Show all work.. 5 x 7 y = x y =. x+ 5 y = x+ y =. x y = 6x+ y =. x 8 y = 6x+ y = 5. 5 x y = 5 7x+ 5y = 8 6. x 5y = x+ y = 5 5
MULTIPLYING POLYNOMIALS Objective: To multiply polynomials Example ( x + )( x + x ) Original problem = ( x + )( x + x ) Distribute the x = ( x + )( x + x ) Distribute the = x + x x + x + 8x 6 Perform multiplication = x + 6x + 5x 6 Combine like terms Multiply the polynomials. Show all work.. ( x + )( x ). ( x 5)( x + 5). ( x + )( x ). ( x )(x + ) 5. (x 5)( x ) 6. ( x + x )( x + ) 7. (x )( x x + ) 8. ( x + )( x x + 5) 9. ( x x + )( x + x ) 6
FACTORING POLYNOMIALS Objective: To factor a polynomial using the greatest common factor (GCF) Example Factor: x + x Find the greatest common factor for both the numbers and the variables x ( x + ) Factor out the GCF x² from each term **Hint: To check your answer simply use the distributive property to check if your answer is the same as the original problem. Factor completely.. x 6x. x + 6xy 8. x y x y. 7 5x 50x 5. 6x 8x + x 6. xy 6y + xy 7. 0x x 5 8x 8. x y 6x y xy + 9. 6y + y 7
FACTORING POLYNOMIALS Objective : To factor the difference of two squares In order for a polynomial to be a perfect square it must meet three conditions:. there are only two terms. each term is a perfect square. it must have a minus sign Example Example Factor: Factor: x x 5 5 x x 9 9 x 5 Find the square root of each factor x 8 Find the square root of each factor ( x + 5)( x 5) Follow the factoring pattern ( x + 9)(x 9) Follow the factoring pattern Objective : To factor trinomials of the form x + bx + c and x + bx c Example Factor: x 0x + 6 x 6 x 8 x Since the coefficient of the x² term is one, list all the factors of your last term (x )(x 8) Solution Next look at the last sign of the trinomial, since it is a plus sign pick the set of factors whose sum gives you the middle term 0. + 8 = 0 therefore and 8 are the correct factors. Since the middle term is 0 you would need and 8. ** Hint: To check your answer, simply use the foil method. If your answer is the same trinomial that you started with, you are correct! Example x Factor: x x 8 x 5x Since the coefficient of the x² term is one, list all the factors of your last term x 6 (x + )(x 8) Solution Next look at the last sign of the trinomial, since it is a minus sign pick the set of factors whose difference gives you the middle term 5 8 = 5 therefore and 8 are the correct factors. Since the middle term is 5 you would need + and 8 ** Hint: To check your answer, simply use the foil method. If your answer is the same trinomial that you started with, you are correct! 8
Factor completely. (Remember to factor out the GCF if there is one.) Show all steps.. x 6. 6 x 00. x x. x + 7x + 7 5. x 5x + 6x 6. x x + 8 7. 8 x 8. x x + 9. x 0. x y + 6xy + 8y. x 00. x 0x 56 9
THE QUADRATIC FORMULA Objective: To use the quadratic formula to solve a quadratic equation ax +bx+c Example x 5x = 0 Original equation a =, b = 5, c = Assign the coefficients to a, b, and c Remember the quadratic formula: ± b b ac a ( 5) ± ( 5) ()( ) () Substitute the values for a, b, and c into the formula 5 ± Simplify 5 + 5 + 5 Separate the solutions 5 Simplify the radicals Simplify the fractions Example x x = Original equation x x = 0 Write equation in standard form a =, b =, c = Assign the coefficients to a, b, and c ( ) ± ( ) ()( ) () Substitute the values for a, b, and c into the formula ± 8 Simplify + 8 8 Separate the solutions + 57 57 Simplify the radicals 7 + 57 7 57 Reduce the fractions 0
Use the quadratic formula to solve each equation. Leave answers in reduced radical form. Show all work.. x + x = 0. x 8x + = 0. x x + 5 = 0. x + 9x = 7 5. 5x + 6. 5x + x =
FACTORING TO SOLVE A QUADRATIC EQUATION Objective: To solve a quadratic equation by factoring and using the Zero-Product property Example x 7x + 0 = 0 Original equation ( x 5)( x ) = 0 Factor x 5 = 0 x = 0 Set each factor equal to zero 5 Solve each equation for x Example 5x + Original equation x 5x = 0 Write equation in standard form by bringing all terms to one side ( x 8)( x + ) = 0 Factor x 8 = 0 x + = 0 Set each factor equal to zero 8 Solve each equation for x Solve the equations by factoring and then use the Zero-Product property. Show all work.. x x 8 = 0. x 6x 6 = 0. x x + 0 = 0. x + 8x= 0 5. 8x 6. 6x 6x = 0 7. x 8 = 0 8. 0x 5 9. x = 7x
SIMPLIFYING RADICALS Objective: To simplify radicals Example 6 Original problem 9 7 Find factors that are perfect squares 9 7 Separate factors 7 Simplify the radical with the perfect square Example 8x Original problem 6 x Find factors that are perfect squares Separate factors 6 x x Simplify the radicals with the perfect squares 8 x Multiply constant terms Hint: When finding the square root of a variable, divide the exponent on the variable by Simplify each radical completely. Assume that each variable represents a positive real number. Show all work.. 5. 98. 80x. 7 5. 6 5 x 6. 6x y 7. x x 5 8. 50y 9. 5 x y 7x y
DATA ANALYSIS Objective : To create a box and whisker plot given a set of data Example Draw the box-and-whisker plot for the following data set: 79 77 87 86 99 87 80 9 You will need to find the following values: a) the minimum and maximum values b) Q, Q, and Q Solution: Start by putting the data in numerical order. 77 79 80 86 87 87 9 99 Find the median, or the middle number. Since there are eight data points, the median will be the average of the two middle values: (86 + 87) = 86.5 The median is also called quartile or Q Next split the list of numbers into two halves: 77 79 80 86 87 87 9 99 Find the median, or middle number of each half. Since the halves of the data set each contain an even number of values, the sub-medians will be the average of the middle two values. Q = (79 + 80) = 79.5 Q = (87 + 9) = 90.5 The minimum value of the data set is 77 and the maximum value is 99, so you have: min = 77, Q = 79.5, Q = 86.5, Q = 90.5, ma 99 Plot each of these five numbers on a number line. The "box" part of the plot goes from Q to Q. 77 79.5 86.5 90.5 99 The "whiskers" part of the graph are drawn to the endpoints.
Objective : To find the mean, median, mode, and range of a stem-and-leaf plot Example Find the mean, median, mode, and range of the following stem-and-leaf plot: The number 8 would be represented as Stem Leaf 8 Stem Math Test Scores (out of 50 pts) 6 8 Leaf 0 5 5 7 8 9 5 0 0 0 Therefore the scores represented in this stem-and-leaf are:, 6, 8, 0,,,, 5, 5, 7, 8, 9, 50, 50, 50 Solution Mean = + 6 + 8 + 0 + + + + 5 + 5 + 7 + 8 + 9 + 50 + 50 + 50 = 660 = 5 5 Median = 5 Mode = 50 Range = 50 = 6 Definitions: Mean the average of all the numbers Median the middle number of a set of numbers in order from smallest to largest Mode the number that appears most often (there can be more than one mode) Range the difference between the largest and smallest numbers Quartile (Q ) the median of the first half of the set of numbers Quartile (Q ) the median of the second half of the set of numbers Interquartile range the difference between Q and Q 5
Use the space provided to create a box and whisker plot for the set of data and answer the questions that follow. Show all work..5 5 7.5 5.5 7.5 8 6 5.5 6.5. mean =. median =. range =. interquartile range = Hint: First put the numbers in order from least to greatest. Construct a Stem and Leaf Plot with the following temperatures for May. Show all work. 76, 80, 70, 8, 68, 58, 57, 6, 70, 6, 67, 6, 65, 7, 7, 8, 55, 79, 78 Find the mean, median, mode, and range for stem-and-leaf plot:. Mean =. Median =. Mode =. Range = 6