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RECORDED CLASSES MATH 301 SLOPE A line connects any two points. How steep is the slope? change in height over change in length Δy divided by Δx proportional relationship of y and x the greater the value the greater the slope positive slope up by Δy right by Δx graph the line lower left to upper right negative slope down by Δy right by Δx graph the line upper left to lower right -5/-3 is a positive slope

INDEX MATH 301 1. Slope 444 2. Slope 1 333 3. Slope 2 333 4. Slope 3 333 5. Direct Variation 333 negative divided by a negative The slope is the same as 5/3. horizontal line like y = 4 slope = 0 vertical line like x = 3 The slope cannot be defined. linear functions have a slope

SLOPE 1 Meaning Clusters Horizontal lines have a slope of zero. The linear equation y = 4 intersects points (1, 4) and (3,4). ^Δy/Δx = ^(4 – 4)/(3 – 1) = ^0/2 = 0 Δy is the difference or change in y. Δxis the difference or change in x. Greater changes in the value of y result in steeper slopes. A slope of ¼ means the line moves 4 units right for every 1 unit up. A slope of 4 means the line moves 1 unit right for every 4 units up. A positive slope moves from lower left to upper right. A slope of -2 means the line moves 1 unit right for every 2 units down. A negative slope moves from lower right to upper left. Steeper lines have greater values. Vertical lines do not have a slope.

The linear equation x = 6 intersects points (6, 1) and (6, 5). ^Δy/Δx = ^(5 – 1)/(6 – 6) = ^4/0 = undefined MEMORY TRIGGERS TESTS Horizontal__________________________ ___________________________________ ____linear__________________________ ___________________________________ _______changes_____________________ ___________________________________ ___slope____________________________ ___________________________________ __________________line______________ ___________________________________ ___positive__________________________ ___________________________________ _________________means_____________ ___________________________________ ____negative________________________ ___________________________________ ____vertical_________________________ ___________________________________ ______________equation______________ ___________________________________ Speed Learning format by Carl Peterson ©2005

SLOPE 2 Meaning Clusters Slope problem: Find another point on a line containing the point (5, 4) with a slope of ^-3/7. x1 = 5 y1 = 4 slope = (y2 – y1) ÷ (x2 – x1) or (y2 – y1) (x2 – x1) ^-3/7 = (y2 – 4) ÷ (x2 – 5) or (y2 – 4) (x2 – 5) Pick a convenient value for x2. x2= 6 ^-3/7 = (y2 – 4) ÷ (6 – 5) or (y2 – 4) (6 – 5) ^-3/7 = (y2 – 4) ÷ 1 Cross multiply. 1(-3) = 7(y2 – 4) Simplify using the distributive property. -3 = 7y2 – 28 -3 + 28 = y2 – 28 + 28 25 = y2 Another point on that line is (6, 25). Speed Learning format by Carl Peterson ©2005

MEMORY TRIGGERS TESTS Slope ______________________________ ____________another ________________ ____(5, 4)__________________________ ___________________________________ slope ______________________________ ___________________________________ ^-3/7 = ______________________________ ___________________________________ ________________________________ x2. ___________________________________ _________________÷_________________ ___________________________________ _________________ multiply __________ ___________________________________ Simplify ___________________________ ___________________________________ ____-3 = ___________________________ ___________________________________ _____________________________(6, 25) ___________________________________

SLOPE 3 Meaning Clusters A positive change in y moves up on a graph. A negative change in y moves down. A positive change in x moves right on a graph. A negative change in x moves left. Slope measures the change in y divided by the change in x. A line contains points (3, 4) and (6, 5). Subtract y1 from y2. 5 – 4 = Δy = 1 Subtract x1 from x2. 6 – 3 = Δx = 3 ^Δy/Δx= ^1/3 The slope moves 1 unit up and 3 units right. Reversing the order of the points does not affect the slope. ^(4 – 5)/(3 – 6) = ^-1/-3 = ^1/3

MEMORY TRIGGERS TESTS ___positive__________________________ ___________________________________ ____negative________________________ ___________________________________ _________change____________________ ___________________________________ _________measures__________________ ___________________________________ ____line____________________________ ___________________________________ Subtract____________________________ ___________________________________ ___________ Δx _____________________ ___________________________________ _____________moves_________________ ___________________________________ Reversing___________________________ ___________________________________ Speed Learning format by Carl Peterson ©2005

DIRECT VARIATION Meaning Clusters y = 3x Equations with two variables and one coefficient are direct equations. y is 3 times larger than x. The coefficient 3 is called the constant of variation. A direct equation is linear as well. Graphing it results in a line. The constant of variation is the slope of the line. The line graphed in y = 3x moves 3 units up and 1 unit right. Every direct variation contains the origin. If x equals 0: y = 3(0) y = 0 If y equals 0: 0 = 3x ^0/3 = ^3x/3 0 = x Anything multiplied by 0 will equal 0. The point (0,0) will always be a solution.

MEMORY TRIGGERS TESTS y =_________________________________ ___________________________________ Equations___________________________ ___________________________________ _____coefficient_____________________ ___________________________________ ___direct___________________________ ___________________________________ _____constant of variation_____________ ___________________________________ _____graphed_______________________ ___________________________________ _____________________contains_______ ___________________________________ _______equals_______________________ ___________________________________ _______multiplied___________________ ___________________________________ ____point___________________________ ___________________________________ Speed Learning format by Carl Peterson ©2005