Relations Math 314 Time Frame Slope Point Slope

Relations Math 314

Time Frame • • Slope Point Slope Parameters Word Problems

Substitution • Sometimes we look at a relationship as a formula • Consider 2 x + 8 y = 16 • We have moved away from a single variable equation to a double variable equation • It cannot be solved as is!

Substitution • • • If we know x = 4 2 x + 8 y = 16 2(4) + 8 y = 16 8 y = 8 y=1

Substitution • We could say that the point x = 4 and y = 1 or (4, 1) satisfies the relationship. • Ex #2. Given the relationship 5 x – 7 y = 210, use proper substitution to find the coordinate (2, y) • (2, y) 5 x – 7 y = 210 • 5(2) – 7 y = 210 • 10 – 7 y = 210 • -7 y = 200 • y = - 28. 57 • (2, -28. 57)

Substitution • Ex. #3: Given the relationship 8 x + 5 y = 80 (x, 8) • (x, 8) 8 x + 5 y = 80 • 8 x + 5(8) = 80 • 8 x + 40 = 80 • 8 x = 40 • x=5 • (5, 8)

Substitution • Ex: #4 Given the relationship y= 3 x 2 – 5 x – 2 • (-3, y) y = 3 (-3)2 – 5 (-3) – 2 • y = 3 (9) + 15 – 2 • y = 40 • (-3, 40) • Stencil #2 (a-j)

Substitution • Given the relationship

Linear Relations • • • We recall… Zero constant relation – horizontal Direct relation – through origin Partial relation – not through origin The characteristic here is the concept of a straight line – a never changing start and where it crosses the y axis

Example • Line A • Line B • We say line A has a more of a slant slope or a steeper slope • (6 compared to 2 is steeper or -6 compared to -2 is steeper).

Variation Relations Name of Relation Formula Direct Relation y = mx Partial Relation y = mx + b Zero Variation y=b Inverse Variation y = m x Graph

Slope • We define the slope • What makes a slope? as the ratio between the rise and the run Rise • Slope = m = rise run Run

Formula for Slope • If we have two points (x 1, y 1) (x 2, y 2) • Slope = m = y 1 – y 2 = x 1 – x 2 • Remember it is Y over X! • Maintain order y 2 – y 1 x 2 – x 1

Slope A (x 1, y 1) B(x 2, y 2) • Consider two points A (5, 4), B (2, 1) what is the slope?

Calculating Slope • Slope = m = y 1 – y 2 = x 1 – x 2 y 2 – y 1 x 2 – x 1 (5, 4) (2, 1) 4 -1 5 -2 3 3 m=1 (x 1, y 1) (x 2, y 2)

Ex # 2 A = (-4, 2) (x 1, y 1) -4 – 2 2 --4 - 6 6 m = -1 B=(2, -4) (x 2, y 2)

Ex #3 (4, 5) (x 1, y 1) (1, 1) (x 2, y 2)

Understanding the Slope • If m or the slope is 2 this means a rise of 2 and a run of 1 (2 can be written as 2 ) 1 • If m = - 5, this means a rise of -5 and right 1 • If m= -2 this means rise of -2 right 3 3 • Rise can go up or down, run must go right

Consider y = 2 x + 3 • What is the slope, y intercept, rise & run? • We can write the slope 2 as a fraction 2 1 • We have a y intercept of 3 • This means rise of 2, run of 1 • Look at previous slide for slope of 4/3

Ex#1: y=2 x+3 Question: Draw this line (1, 5) Where can you plot the y intercept? 0, 3 What is the y intercept? What is the slope What does the slope mean? Up 2, Right 1

Example • What do you think the slope will be; calculate it. (-4, 2) (2, 2) If a line//x-axis slope = 0

Example (2, 2) zero! If a line // y-axis: slope is undefined (2, -3)

In Search of the Equation • We have seen that the linear relation or function is defined by two main characteristics or parameters • A parameter are characteristics or how we describe something • If we consider humans, a parameter would be gender. (We have males & females). There can be many other parameters (blonde hair, blue eyes, etc. )

In Search of the Equation Notes • The parameters we are concerned with are… • Slope = m = the slope of the line • y intercept = b = where the line touches or crosses the y axis (It can always be found by replacing x = 0) • x intercept = where on the graph the line touches or crosses the x axis. (let y = 0)

In Search of the Equation Notes • We stated in standard form the equation for all linear functions by y = mx + b. Recall… • y is the Dependent Variable (DV) • m is the slope • x is the Independent Variable (IV) • b is the y intercept parameter • The key is going to be finding the specific parameters.

General Form • You will also be asked to write in general form • General Form Ax + By + C = 0 • A must be positive • Maintain order x, y, number = 0 • No fractions

General Form Practice • Consider y = 6 x – 56 • -6 x + y + 56 = 0 • 6 x – y – 56 = 0

Standard & General Form Example #1 • State the equation in standard and general form. • Consider find the equation of the linear function with slope of m and passing through (x, y). • m = -6 (-2, -3) • (-2, -3) -3 = -6 (-2) + b • -3 = 12 + b • -15 = b • b = -15

Example #1 Solution Con’t • y = -6 x – 15 (Standard) • Now put this in general form • 6 x + y +15 = 0 (General)

Standard & General Form Ex. #2 • m = -2 (5, - 3) 3 • -3 = (-2) (5) + b 3 • -3 = -10 + b 3 • -9 = -10 + 3 b • 1 = 3 b • b = 1/3 • y = -2 x + 1 (SF) 3 3 • Now General form • Get rid of the fractions; how? Given y = -2 x + 1 3 3… Anything times the bottom gives you the top • 3 y = -2 x + 1 • 2 x + 3 y – 1 = 0

Standard and General Form Ex #3 • m=4 5 (-1, -1) • -1 = 4 x + b 5 • -5 y = -4 x + 5 b • 5 (-1) = 4 (-1) + 5 b • -5 = -4 + 5 b • -1 = 5 b • b = -1/5 • y = 4 x – 1 5 5 • 5 x – 1/5 (standard form) • 5 y = 4 x – 1 • -4 x +5 y + 1 = 0 • 4 x – 5 y – 1 = 0 (general form)

The Point Slope Method Con’t • Consider, find the equation of the linear function with slope 6 and passing through (9 – 2). • Take a look at what we know based on this question. • m=6 • x=9 • y = -2

Finding the Equation in Standard Form • We know y = mx + b • We already know y = 6 x + b • What we do not know is the b parameter or the y intercept • We will substitute the point • (9, -2) - 2 = (6) (9) + b • -2 = 54 + b • -56 = b • b = - 56 • y = 6 x – 56 (this is Standard Form) • Standard from is always y = mx + b (the + b part can be negative… ). You must have the y = on the left hand sides and everything else on the right hand side.

General Form • In standard form y = 6 x – 56 • In general form -6 x + y + 56 = 0 • 6 x – y – 56 = 0

Example #1 8 a on Stencil • In the following situations, identify the dependent and independent variables and state the linear relations • Little Billy rents a car for five days and pays $287. 98. Little Sally rents a car for 26 days and pays $1195. 39. • D. V $ Money $ • I. V. # of days

Example #1 Soln Con’t • Try and figure out the equation • y = mx + b (you want 1 unknown) Unknown • (5, 287. 98) (26, 1195. 39) • m = (287. 98 – 1195. 39) 5 – 26 • m = 43. 21

Example #1 Soln Con’t • • Solve for b… y = mx + b (5, 287. 98) 287. 98 = 43. 21 (5) + b 287. 98 = 216. 05 + b 71. 93 = b b = 71. 93 y = 43. 21 x + 71. 93

Example #2 8 b on Stencil • A company charges $62. 25 per day plus a fixed cost to rent equipment. Little Billy pays $1264. 92 for 19 days. • I. V. # of days • D. V. Money • m = 62. 25

Example #2 8 a Soln • • • y = mx + b (19, 1264. 92) 1264. 92 = 62. 25 (19) + b 1264. 92 = 1182. 75 + b 82. 17 = b b = 82. 17 y = 62. 25 x + 82. 17

Solutions 8 c, d, e • • • 8 c) IV # of days; DV $ y = 47. 15 x + 97. 79 8 d) IV # of days; DV $ y = 89. 97 x + 35. 22 8 e) IV # of days DV $ y= 45. 13 x + 92. 16

Homework Help • What is the value of x given • 3=1+1 4 2 x • Eventually, x on the left side, number on the right side • 3– 1=1 4 2 x • 6 x – 4 x = 8 • Important step to • -2 x = 8 understand • x = -4

Homework Help • What is the opposite of ½ ? • Answer is –½ • If asked what is the opposite of subtracting two fractions… i. e. ¼ - ½ , find the answer (lowest common denominator and then reverse the sign. • When told price increases 10% each year… calculate new price after year 1 and then multiply that number by. 1 again to calculate price increase for year 2. For example, you have $100 and increases 10%. After year 1 $110 (100 x. 1 + 100) & after year two $121 (110 x. 1 + 110).