relations math 314 time frame slope point slope parameters word problems
TRANSCRIPT
RelationsRelations
Math 314Math 314
Time FrameTime Frame
• Slope
• Point Slope
• Parameters
• Word Problems
SubstitutionSubstitution
• Sometimes we look at a relationship as a formula
• Consider 2x + 8y = 16
• We have moved away from a single variable equation to a double variable equation
• It cannot be solved as is!
SubstitutionSubstitution
• If we know x = 4
• 2x + 8y = 16
• 2(4) + 8y = 16
• 8 + 8y = 16
• 8y = 8
• y = 1
Substitution Substitution • We could say that the point x = 4 and y = 1 or
(4,1) satisfies the relationship. • Ex #2. Given the relationship 5x – 7y = 210, use
proper substitution to find the coordinate (2,y)• (2,y) 5x – 7y = 210• 5(2) – 7y = 210• 10 – 7y = 210• -7y = 200• y = - 28.57• (2, -28.57)
Substitution Substitution
• Ex. #3: Given the relationship 8x + 5y = 80 (x,8)
• (x,8) 8x + 5y = 80• 8x + 5(8) = 80• 8x + 40 = 80• 8x = 40• x = 5 • (5,8)
Substitution Substitution
• Ex: #4 Given the relationship y= 3x2 – 5x – 2
• (-3,y) • (-3,y) y = 3 (-3)2 – 5 (-3) – 2• y = 3 (9) + 15 – 2• y = 40• (-3,40) • Stencil #2 (a-j)
Substitution Substitution
• Given the relationship
Linear RelationsLinear 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 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 RelationsVariation Relations
Name of Relation Formula Graph
Direct Relation y = mx
Partial Relation y = mx + b
Zero Variation y = b
Inverse Variation y = m
x
SlopeSlope
• What makes a slope?
Rise
Run
• We define the slope as the ratio between the rise and the run
• Slope = m = rise
run
Formula for SlopeFormula for Slope
• If we have two points (x1, y1) (x2, y2)
• Slope = m = y1 – y2 = y2 – y1
x1 – x2 x2 – x1
• Remember it is Y over X!
• Maintain order
1 2 3 4 5 6 7 8-1-2
1
2
3
4
5
-1
-2
-3
-4
-5
x
y
A (x1, y1)
B(x2, y2)
SlopeSlope
• Consider two points
A (5,4), B (2, 1) what is the slope?
Calculating SlopeCalculating Slope
• Slope = m = y1 – y2 = y2 – y1
x1 – x2 x2 – x1
(5, 4) (2, 1) 4 - 1 5 - 2 3 3m = 1
(x1,y1) (x2,y2)
Ex # 2Ex # 2 A = (-4, 2) B=(2, -4)
(x1,y1) (x2,y2) -4 – 2
2 - - 4
- 6
6
m = -1
1 2 3 4 5-1-2-3-4-5
1
2
3
4
5
-1
-2
-3
-4
-5
x
y (4, 5)
(1, 1)
3
414
15slope
Ex #3Ex #3
(x2,y2)
(x1,y1)
21
21
xx
yyslope
Understanding the SlopeUnderstanding 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 = 2x + 3Consider y = 2x + 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
1 2 3 4 5-1-2-3-4-5
1
2
3
4
5
-1
-2
-3
-4
-5
x
y
Ex#1: y=2x+3Ex#1: y=2x+3
0,3
(1,5)
Question: Draw this line
What is the y intercept?
What is the slope
What does the slope mean?
Where can you plot the y intercept?
Up 2, Right 1
1 2 3 4 5-1-2-3-4-5
1
2
3
4
5
-1
-2
-3
-4
-5
x
y
(-4, 2) (2,2)
06
0
)4(2
22
slopeIf a line//If a line//xx--
axisaxis
slope = 0slope = 0
ExampleExample
• What do you think the slope will be; calculate it.
1 2 3 4 5-1-2-3-4-5
1
2
3
4
5
-1
-2
-3
-4
-5
x
y
(2,-3)
(2,2)0
522
32
slope
If a line // If a line // yy--axis:axis:
slope is slope is undefinedundefined
ExampleExample
zero!
In Search of the EquationIn 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 NotesIn 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 NotesIn 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 FormGeneral 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 PracticeGeneral Form Practice
• Consider y = 6x – 56
• -6x + y + 56 = 0
• 6x – y – 56 = 0
Standard & General FormStandard & General FormExample #1Example #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’tExample #1 Solution Con’t
• y = -6x – 15 (Standard)
• Now put this in general form
• 6x + y +15 = 0 (General)
Standard & General Form Ex. #2Standard & General Form Ex. #2• m = -2 (5, - 3) 3 • -3 = (-2) (5) + b 3• -3 = -10 + b 3• -9 = -10 + 3b• 1 = 3b• 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
• 3y = -2x + 1• 2x + 3y – 1 = 0
Standard and General Form Ex #3 Standard and General Form Ex #3
• m = 4
5 (-1, -1)• -1 = 4 x + b
5• -5y = -4x + 5b• 5 (-1) = 4 (-1) + 5b• -5 = -4 + 5b • -1 = 5b • b = -1/5
• y = 4x – 1
5 5• 5 x – 1/5 (standard
form)• 5y = 4x – 1• -4x +5y + 1 = 0 • 4x – 5y – 1 = 0
(general form)
The Point Slope Method Con’tThe 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 Finding the Equation in Standard FormForm
• We know y = mx + b• We already know y = 6x + 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 = 6x – 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 FormGeneral Form
• In standard form y = 6x – 56
• In general form -6x + y + 56 = 0
• 6x – y – 56 = 0
Example #1 8a on StencilExample #1 8a 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’tExample #1 Soln Con’t
• Try and figure out the equation
• y = mx + b (you want 1 unknown)
• (5, 287.98) (26, 1195.39)
• m = (287.98 – 1195.39)
5 – 26
• m = 43.21
Unknown Unknown
Example #1 Soln Con’tExample #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 = 43.21x + 71.93
Example #2 8 b on StencilExample #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 8a Soln Example #2 8a 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 = 62.25x + 82.17
Solutions 8 c, d, eSolutions 8 c, d, e
• 8c) IV # of days; DV $
• y = 47.15x + 97.79
• 8d) IV # of days; DV $
• y = 89.97x + 35.22
• 8e) IV # of days DV $
• y= 45.13x + 92.16
Homework HelpHomework 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• 6x – 4x = 8• -2x = 8• x = -4
• Important step to understand
Homework HelpHomework 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).