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Linear Functions Slope Parallel & Perpendicular Lines Different Forms of Linear Equations $100 $100 $100 $100 $200 $200 $200 $200 $300 $300 $300 Solving Equations and Inequalities Review Misc. Linear Functions $100 $100 $200 $200 $300 $300 $300 $400 $400 $400 $400 $400 $400 $500 $500 $500 $500 $500 $500 $100 Question Linear Functions Explain how you know if a graph represents the following: Function: Linear function: Direct Variation (Proportional Relationship): $100 Answer Linear Functions Function: passes VLT Linear function: passes VLT & straight line Direct Variation (proportional relationship: passes VLT, straight line & crosses through (0,0) $200 Question Linear Functions Which table(s) are linear? Explain how you know. A. B. $200 Answer Linear Functions B.) x and y are both going up with constant rates * same rate of change * Rate of change = change of y change of x Table A – not a constant rate of change $300 Question Linear Functions Which of the following equations is not linear? Explain or show how you know. A) y = 2x2 – 7 B) -6 = y C) 4x – 2y = 10 D) y = 3x + 1 $300 Answer Linear Functions A) y = 2x2 – 7 not linear--exponent B) -6 = y horizontal—straight line C) 4x – 2y = 10 standard form x and y-int. --line D) y = 3x + 1 slope-int.—always form a line $400 Question Linear Functions Explain if each equation represents a direct variation (proportional relationship). Identify the constant of variation if it is a direct variation. A.) 3y = 4x + 1 B.) y + 3x = 0 $400 Answer Linear Functions Solve each equation for y. Look to see if it’s in y = kx form k = constant of variation A.) 3y = 4x + 1 y = 4/3x + 1 Not in y = kx form (+1) B.) y + 3x = 0 y = -3x Yes, in y = kx form k = m -3 $500 Question Linear Functions Let y = 2x + 9. If the value of x increases by 6, which of the following best describes the change in the value of y. a.) decreases by 6 b.) increases by 6 c.) increases by 12 d.) increases by 21 $500 Answer Linear Functions As x goes up by 6 The value of y. c.) increases by 12 $100 Question Slope Find the slope of the line (0, 4) (3, -5) $100 Answer Slope Rise Run m = -3 $200 Question Slope Write the equation of the line that passes through each pair of points in slopeintercept form (4,6) and (2, 2) $200 Answer Slope m=2 Remember a neg. divided by a neg. is positive 1. Label Points and use slope formula 2. Choose a point and use point-slope formula y – 6 = 2(x – 4) Distribute y – 6 = 2x – 8 Add 4 to both sides y = 2x - 2 $300 Question Slope Put the following equation into slopeintercept form. Identify the slope and yintercept. Then use the slope and y-int. to graph the line. 3x – y = 2 $300 Answer Slope 3x – y = 2 -3x -3x -y = -3x + 2 -1 -1 -1 y = 3x -2 m = 3 b = -2 $400 Question Slope Laurel graphed the equation y = -2x + 5. Katelyn then graphed an equation that was a line that was not as steep as Laurel’s. Which equation could have been the one Katelyn graphed? a.) y = -3x + 5 b.) y = 1/2x + 6 c.) y = 4x – 2 d.) y = -2x + 3 $400 Answer Slope B.) y = 1/2x + 6 is not as steep. Fractions (between -1 and 1: non-improper) are less steep than any integer— even if it’s negative. $500 Question Slope The cost of hiring Zach as a painter is given by the linear equation C = 10h + 100, where h is the number of hours Zach works. Identify the slope and y-int. What does the slope of the line represent? What does the y-intercept represent? $500 Answer Slope m = 10 The slope means Zach earns $10 per hour. b = 100 The y-intercept represents base charge of hiring Zach $100 Question Parallel & Perpendicular Lines What are two different ways that lines can be perpendicular? $100 Answer Parallel & Perpendicular Lines Vertical lines are perpendicular to a horizontal lines. Ex. x = 3 and y = -2 When the product of slopes = -1 Ex. 4 and -1/4 $200 Question Parallel & Perpendicular Lines A line has the equation x + 2y = 5 What is the slope of a line parallel to this line? a.) – 2 b.) - ½ c.) ½ d.) 2 $200 Answer Parallel & Perpendicular Lines A line has the equation x + 2y = 5 1. Put line in slope-int. form y = -1x + 5 2 2 2. Parallel -- same slope -- b.) - ½ $300 Question Parallel & Perpendicular Lines $300 Answer Parallel & Perpendicular Lines A. 1 and -1 are “opposite reciprocals” $400 Question Parallel & Perpendicular Lines $400 Answer Parallel & Perpendicular Lines Line AB has a slope of 1 and Line BC has a slope of -3/2 and Line AC has a slope of 0. None of the slopes will have a product of -1 so D is the answer $500 Question Parallel & Perpendicular Lines Write an equation that is perpendicular to the given line below that passes through the point (- 6, 2) $500 Answer Parallel & Perpendicular Lines 1. Slope will be -3 (opp. reciprocal) 2. Use point-slope form y- 2 = -3(x – (-6)) Distributive Prop. y = -3x -16 $100 Question Different Forms of Linear Equations Find and use the x and y intercepts to graph the line. -x + 3y = 6 $100 Answer Different Forms of Linear Equations -x + 3y = 6 -x + 3y = 6 0 + 3y = 6 -x + 3(0) = 6 y-int. = 2 -x = 6 (0,2) x-int. = -6 (-6,0) (0,2) (-6,0) $200 Question Different Forms of Linear Equations Find and use the x and y intercepts to graph the line. -2x = 12 + 4y $200 Answer Different Forms of Linear Equations -2x = 12 + 4y -4y from both sides -2x - 4y = 12 Now in standard form x-int. = (-6,0) y-int. = (0,-3) (-6,0) (0, -3) $300 Question Different Forms of Linear Equations What is the x-intercept of the linear function f(x) = -3x + 6? Note: f(x) is dependent variable (y) a.) -2 b.) 2 c.) 3 d.) 6 $300 Answer Different Forms of Linear Equations f(x) = -3x + 6 Think: y = -3x + 6 add 3x to both sides –standard form y + 3x = 6 0 + 3x = 6 x-int. = 2 (b) $400 Question Different Forms of Linear Equations A line has a slope of and passes through the point (-3, 4). What is the equation of the line in point-slope form? What is the equation of the line in slope-intercept form? $400 Answer Different Forms of Linear Equations Point-slope form y- 4 = [x – (-3)] y–4= (x + 3) Slope-Intercept Form y = x+6 $500 Question Different Forms of Linear Equations Is every linear relationship a direct variation? Is every direct variation/proportional relationship a linear relationship? Explain. $500 Answer Different Forms of Linear Equations Every linear relationship is not a direct variation—only if the y-int. is 0. However, every direct variation is linear because it has a constant rate of change. Direct Variation/prop. relat.: y = 3x (also linear) Not a direct variation y = 3x +5 (is linear) $100 Question Solving Equations and Inequalities Review Solve and Graph -2m < 8 $100 Answer Solving Equations and Inequalities Review -2m < 8 - 2 -2 m > -4 Open to the right $200 Question Solving Equations and Inequalities Review Solve and Graph 4a – 10 > 6a $200 Answer Solving Equations and Inequalities Review 4a – 10 > 6a -10 > 2a -5 > a Open to the left 4 y 8y 2 2 7 7 $300 Question Solving Equations and Inequalities Review Solve and Graph $300 Answer Solving Equations and Inequalities Review Closed to the right $400 Question Solving Equations and Inequalities Review Solve. 2(t - 4) = 3 – (3t + 1) $400 Answer Solving Equations and Inequalities Review 2(t - 4) = 3 – (3t + 1) 2t – 8 = 3 – 3t – 1 2t – 8 = 2 – 3t 5t = 10 t=2 $500 Question Solving Equations and Inequalities Review Janine has job offers at two companies. One company offers a starting salary of $28,000 with a raise of $3000 each year. The other company offers a starting salary of $36,000 with a raise of $2000 each year. Find when the two offers are the same. Which offer will be better if you plan to stay longer than 10 years? $500 Answer Solving Equations and Inequalities Review $28,000 + $3000x = $36,000 + $2000x Solve for x. x = 8 years. They will both make $52,000 After 8 years, the starting rate of $28,000 with a increase of $3000 per year will earn more $. $100 Question Miscellaneous The table shows an employee’s pay per hour. Determine if there is a direct variation between the pay and number of hours worked. If so, find the equation of direct variation $100 Answer Miscellaneous You can use the ratio to check. 17/2 = 8.5 and 34/4 = 8.5 ratios are = (proportional) so it’s a direct variation The equation would be y = 8.5x $200 Question Miscellaneous Make a table and graph for y = -x + 3 Is this equation linear? Explain. $200 Answer Miscellaneous Graph has a negative slope—straight line It is a linear function because table has a constant rate of change and graph is a straight line $300 Question Miscellaneous Write the equation of the line that passes through each pair of points in slopeintercept form (-1, 5) and (2, -4) $300 Answer Miscellaneous 1.) Find Slope m = -3 2. ) Choose a point (-1,5) Insert into slope-int. and solve for b OR Use point-slope form then solve for y Y = -3x + 2 $400 Question Miscellaneous Also, explain what each intercept represents $400 Answer Miscellaneous x-intercept (8,0) Namia has enough money to buy 8 lg. popcorns and 0 drinks 4-intercept (0,20) Namia has enough money to buy 20 small drinks and 0 popcorns $500 Question Miscellaneous Prove that ABCD is a parallelogram $500 Answer Miscellaneous ABCD is a parallelogram because opposite sides are parallel. Line AB and Line DC both have a slope of -2/3 and Line AC and Line DC both have an undefined slope (vertical lines) Final Jeopardy Write an equation in slope-intercept form for the given table. Explain what the slope and y-intercept form represent. Then find the cost for 60 additional minutes. Find the rate of change for slope m = 2/5 or 0.4 (0.40 for each add. min.) Pick a point and use slope formula or solve for b to get slope intercept form y = 0.4x + 40 y –int. means it cost $40 with 0 add. min. y = 0.4(60) + 40 It will cost $64 for 60 additional minutes.