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Category 3 STAAR Review – 2017 •Day 6 Essential Vocabulary • Independent • Dependent • Variable • Equation • Table • Inequality • Mapping • Graph y = 3x + 5 y = 3x + 5 y = 3x + 5 y = 3x + 5 y < 3x + 5 X Y -1 2 0 5 1 8 2 11 -1 0 1 2 2 5 8 11 Domain and Range • Domain is the set of all x values • Range is the set of all y values • To find domain: examine the right and left boundaries of the function • To find range: examine the top and bottom boundaries of the function • Whenever a function has two boundaries, both signs should be less than (< or ≤). 2, Ab2B Definition a. Specific points b. Linear c. Quadratic d. Exponential Example a. {(3, 6), (2, 8), (5, 3)} b. y = 3x + 2 c. y = x2 + 5x + 4 d. y = 2(3)x Domain All the possible x-coordinates. a. { 3, 2, 5} b. -∞ < x < ∞ (all real numbers) c. -∞ < x < ∞ d. -∞ < x < ∞ Range All the possible y-coordinates. a. { 6, 8, 3} b. -∞ < y < ∞ c. y ≥ 4 d. y ≥ 0 Domain and Range Graphs • The domain of a function is the set of all the xcoordinates in the functions’ graph Domain 3 ≤ x ≤ 12 The range of a function is the set of all the y-coordinates in the functions’ graph Range 6 ≤ y ≤ 12 Domain and Range of a Graph What is the domain of this function? What is the range of this function? Domain is 0 ≤ x ≤ 4 Range is 1 ≤ y ≤ 5 The average daily high temperature for the month of May is represented by the function t = 0.2n + 80 Where n is the date of the month. (May has 31 days.) n t Min. days in month 0 80 Min. temp. Max. days in month 31 86.2 Max. temp. What is a reasonable estimate of the domain? Answer: 0 ≤ n ≤ 31 What is a reasonable estimate of the range Answer: 80 ≤ t ≤ 87 Put equation in calculator and knowing minimum and maximum days in a month (n), find the min & max for t. Discrete & Continuous Data • Discrete Function has a graph with isolated points. • Domain: x = 1, 2, 3, 4, 5 Cost of renting DVD’s -Can’t have parts of DVD’s. • Continuous Function has a graph that is unbroken. • Domain: x > 0 Face length & ear length is measured – can have fractions. Parallel & Perpendicular Lines • Parallel Lines have equal slope (m) y = ¼ x – 3 and 𝟏 𝟒 y=¼x+6 m = for both • Perpendicular Lines have opposite reciprocal slope (m) y = ¼ x – 5 and y = -4x + 15 m= 𝟏 𝟒 & m = -4 • Lines with the same y intercept and same slope are the same line. y = ¾ x – 9 and y = ¾ x – 9 m = ¾ b = -9 3, Ac2C Parallel and Perpendicular Lines 6 5 4 3 2 1 • Parallel Lines • have the same slope (m) -6 -5 -4 -3 -2 -1 Rise m Run • Perpendicular Lines • have opposite reciprocal slopes 1 2 2 1 -6 -5 -4 -3 -2 -1 7, G.07B y 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6 1 2 1 2 x 1 2 y x 1 2 3 4 5 6 2 1 What is the equation of the line parallel to y = 3x + 2 and passes through the point (1,8). Parallel lines have same slope Slope of new line is 3 Use point slope form (y –y1) = m(x – x1) (y – 8) = 3(x – 1) y – 8 = 3x – 3 ANSWER y = 3x + 5 What is the equation of the line perpendicular to y = -4x + 2 and passes through the point (2,-3) Perpendicular lines have opposite reciprocal slopes (opposite & flip) Slope of new line is 1 4 Use point slope form (y –y1) = m(x – x1) 1 (y + 3) = (x – 2) y+3 ANSWER 4 1 = x 4 1 4 – 1 2 y= x-3 1 2 • What is the slope of a line parallel to the x axis? • Write the equation of the line. M=0 Y=3 • What is the slope of a line perpendicular to the x axis? • Write the equation of the line. • What is the slope of a line parallel to the y axis? • Write the equation of the line. M = undefined X=5 M = undefined X=7 • What is the slope of a line perpendicular to the y axis? • Write the equation of the line. M=0 Y=4 •Day 7 Direct Variation • To find the constant of variation use a linear function (y = kx) and find the slope • The slope, m, is the same thing as k • Example: If y varies directly with x and y = 6 when x = 2, what is the constant of variation? y = kx 6 = k(2) 2 2 3=k The equation for this situation would be y = 3x Direct Variation • Direct Variation ALWAYS goes through the origin, the point (0,0). • Direct variation works the same as a proportions. Important Vocabulary • Equation Must have an = sign • Inequality Must have <, >, ≤, ≥ • Substitution To replace a variable with a number • Inverse Operations • System of Equations • Solution • Perimeter Method of solving an equation More than 1 equation Answer to an equation The distance around an object Perimeter Rectangle = 2l + 2w 4, Ac4B Various Forms of a Linear Equations y mx b Slope-Intercept Form m slope of the line b y intercept Ax By C Standard Form A, B, and C are integers A 0, A must be postive y y1 m x x1 Point-Slope Form m slope of the line x1 , y1 is any point Writing Equations and Inequalities • Identify if the situation warrants an equation (=) or an inequality (<, >, ≤, ≥). • Equations are used when quantities are equal. • Inequalities are used when quantities are not equal. Look for words like: 4, Ac3A No more than No less than At least At most Writing Equations from Graph m = slope of line b = y-intercept Y = mx + b 2 y x4 3 Rise 2 Run 3 Write equations Using Regressions • To find the equation of a function when you are given the table, use the feature of your graphing calculator. Enter the table into the calculator using L1 for x and L2 for y. Then return to function type. and arrow over to CALC and choose the appropriate Press Enter to view equation. 1, Ab1B Writing Equations from Tables • When given a table of values, USE STAT! • Example: What equation describes the relationship between the total cost, c, and the number of books, b? b c 10 75 15 100 20 125 25 150 Answer: c = 5b + 25 3, Ac1C Linear or Quadratic or Exponential • How do I know what type of function to use? • STAAR questions will be either • Linear: LinReg, ax+b • Quadratic: QuadReg, ax2 + bc + c • Exponential: ExpReg, a(b)x • If you aren’t sure, look at the answers and see if they are linear (y = x) or quadratic (y = x²) or exponential (y = abx) Writing Equations from Points • Make a table • USE STAT • Example: Which equation represents the line that passes through the points (3, -1) and (-3, -3)? x 3 -3 1 Answer: y x 2 3 y -1 -3 3, Ac2D Equations that are in Standard Form • Sometimes your equations won’t be in y = mx + b form. • They will be in standard form: Ax + By = C • You must convert them to use the calculator! Example: 3x + 2y = 12 -3x -3x 2y = -3x + 12 2 2 2 3 y x6 2 3, A.05C Step 1: Move the x Step 2: Divide everything by the number in front of y Writing Equations Given Slope and a Point Given a point (-3,7) and m = -2, write the equation of the line. y y1 m(x x1) y 7 2(x (3)) y 7 2(x 3) What if the directions said to write the equation in slope-intercept form? Point Slope Form This is the equation of the line in point-slope form!! y 7 2x 6 +7 +7 y 2x 1 This is the equation of the line in slope intercept form!! Write Equations from Situation • Example: Identify the situation that best represents the amount f(n) = 425 + 50n. Slope (rate of change) = Y intercept (initial value) = 50 425 Find an answer that has: 425 as a non-changing value and 50 as a recurring charge every month, every year, etc… Something like Joe has $425 in his savings account and he adds $50 every month. • Example: Carmen receives a $50 gift card to the local movie theater. Each movie she watches costs $6.75. Which table best describes b, the balance on her gift card after she watches m movies? A 1. Write an equation b = 50 – 6.75m 2. Put equation in y= C 1, Ab1D m b m b 0 0 0 50 1 6.75 1 56.75 5 33.75 5 83.75 7 47.25 7 97.25 m b m b 1 43.25 0 50 2 36.50 1 43.25 4 29.75 5 16.25 7 23 7 2.75 B D Solving Equations • TO SOLVE EVERY EQUATION: (with one variable) • Ask yourself this EVERY time you solve an equation. Should I….. • • • • • • • 1. Do Distributive Property. 2. Combine Like Terms. 3. Move all variables to one side. 4. Add/subtract on both sides. 5. Multiply/divide on both sides last. ***Goal – to get x by itself 6. Check the answer for reasonableness and accuracy. Solving Equations • Example: Mr. Jones charges $25 for an estimate plus $18 per hour to repair dents in car doors. The total charge for a repair is $151. How many hours did Mr. Jones work on the repair? 151 = 18h + 25 A. 5 hours B. 6 hours C. 7 hours D. 8 hours 4, Ac3B 1. Write the equation 2. Substitute each answer for “h” Solving Inequalities • Convert inequalities from Standard form (Ax + By > C) to slope-intercept form (y = mx + b). • Use the same steps as you would for an equation, but remember that if you multiply or divide by a negative number, you must flip the inequality sign! • Example: 4x – 2y ≤ 5 - 4x - 4x -2y ≤ -4x + 5 Because you -2 -2 -2 divided by a negative, you must flip the ≤ to ! y 2x – 𝟓 𝟐 •Day 8 Writing Systems of Equations • Most systems are comprised of 2 types of equations • A total equation that represents the total number of items • And a comparison equation that represents the relationship between the two variables • Identify what the variables represent • Identify which numbers go with each equation type. Writing Systems of Equations • Example: Claudia purchased 12 shirts and jeans for the school year. Jeans cost $22 and shirts cost $15. If Claudia spent a total of $215, write a system of equations that could be used to find the number of shirts that Claudia purchased. j = jeans s = shirts Total Equation Comparison Equation (money) j + s = 12 22j + 15s = 215 Writing Systems of Equations • When rectangles enter the picture: • Remember: Perimeter = 2l + 2w • Example: The length of a rectangle is three times the width. The perimeter of the rectangle is 16 inches. Which system of equations can be used to determine the dimensions of the rectangle? A. l = 3 + w 2l + 2w = 16 C. l = 3w l + w = 16 B. l = 3w 2l + 2w = 16 D. l = w – 3 l + w = 16 Write Linear System Given a Table • Use STAT to enter the first two columns from the given table. • Find the equation for y1 • Use STAT to enter the first and last columns from the given table. • Find the equation for y2 • These two equations are the system. • Notice the solution to the system is in the table – the point (6,7) Write Linear System given a Graph m=3 b = -2 y = mx + b y = 3x - 2 The system is: Y = 3x – 2 Y = - 3/4x + 3 m = -3/4 b=3 y = mx + b y = - 3/4 x + 3 Solving Systems of Equations • The solution to a system of linear equations is the point where the two lines intersect. • If you are unsure how to solve a problem by substitution, elimination, or graphing, you can substitute each answer choice into the equations to see which one works for BOTH equations. Solving Systems by Graphs Solving a System with the graphing calculator • • • • • • • • • • Put equations in Y = mx + b form. Enter first equations into Y1= Enter second equation into Y2= Press Graph (make sure you can see the intersection – if not, press Zoom – Zoom Out Enter, Enter) Press CALC (2nd – Trace) #5 : Intersect Enter Enter Enter Intersection: X= and Y= To solve a system graphically – equations must be in slope intercept form (y = mx + b) Solve the system: y=x+1 Y = 2x + 4 Solution: (-3, -2) Solving a Linear System by Substitution Solving Systems by Elimination Solving Systems by Elimination (2, 2) When to use which method…… Solve the system Write Linear Inequality Given a Graph