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M2-1.1 Representations of Proportional Relationships

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Examples: Notes:

EQ: How can you represent and compare proportional relationships using graphs, tables, and equations?

Proportional Relationship: the ratio of the inputs to the outputs is constant.• Also known as Direct Variation• Written as the equation: y = kx• x represents input value (Independent) • y represents output value (Dependent)• k represents constant- not equal to 0

> k = Constant of Proportionality– AKA... the unit rate

y = kx

= yk xunit rate

where x = 1

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M2-1.1 Representations of Proportional Relationships

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Examples: Notes:

EQ: How can you represent and compare proportional relationships using graphs, tables, and equations?

Graphs that have a proportional relationship:• Straight Line• Go through the origin (0, 0)

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M2-1.1 Representations of Proportional Relationships

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Examples: Notes:

EQ: How can you represent and compare proportional relationships using graphs, tables, and equations?

Ratio: y = kx

xxk=y

x

xy

k = the constant of proportionality = the y-value when x- value = 1 (unit rate)

k = a point on a graph with a ratio of

x = must be 1 to represent the unit ratexy

= k = Rate of Change

xky=

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M2-1.1 Representations of Proportional Relationships

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Examples: Notes:

EQ: How can you represent and compare proportional relationships using graphs, tables, and equations?

y = kx

xxk=y

x

In a linear relationship, any change in an independent variable (x) will produce a corresponding change in the dependent variable (y).

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M2-1.2 Use Similar Triangles to Describe Steepness of a Line

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Examples: Notes:

EQ: How can you connect all of those concepts to describe the steepness of a line?

y = kx: k is the coefficient of x, • k represents the rate of change, constant

of proportionality & unit rate• x = independent quantity• y = dependent quantity

y = kx

xyk =

��������������������

��������

�����������

��������

���������

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M2-1.2 Use Similar Triangles to Describe Steepness of a Line

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Examples: Notes:

EQ: How can you connect all of those concepts to describe the steepness of a line?

Rate of Change: Describes the amount that dependent variable (y) changes compared with amount that the independent (x) variable changes. 1. Choose 2 points on the graph.

> change in y

2. Subtract the y-coordinates of both points (numerator)

3. Subtract the x-coordinates of both points (denominator)

4. Write as a fraction.5. This is your constant= k

change in x

(6, 8)

(3, 4)

8 - 46 - 3

= 43

y = 4 x3

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M2-1.2 Use Similar Triangles to Describe Steepness of a Line

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Examples: Notes:

EQ: How can you connect all of those concepts to describe the steepness of a line?

Slope: (Rate of Change) • y = mx (same as y = kx) represents a

proportional relationship where every point (x, y) on the graph of a line with slope m that passes through the origin (0, 0).

• Describes the direction & steepness of a line

• Represents the ratio of the change in vertical distance (RISE) y to the change in horizontal distance (RUN) x between any 2 points on the line.

• Slope of the line is constant between any two points on the line. 6

14 - (-2) =

3 - (-1) =

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M2-1.2 Use Similar Triangles to Describe Steepness of a Line

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Examples: Notes:

EQ: How can you connect all of those concepts to describe the steepness of a line?

• Slope of a line represents steepness & direction.

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M2-1.2 Use Similar Triangles to Describe Steepness of a Line

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Examples: Notes:y = mx + b• m= change in y (dependent)

• b = translation of the line vertically & where it crosses the y axis. > b = 1 because the actual point is (0, 1)

change in x (independent)

y = mx + b, • where b is NOT equal to zero, • represents a NON-proportional

relationship where every point (x, y) on the graph of a line with slope m that passes through the point (0, b)

• b represents a point on the y-axis

EQ: How can you connect all of those concepts to describe the steepness of a line?

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M2-1.4 Transformation of Lines

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Examples: Notes:

EQ: How can you use knowledge about geometric transformations to transform the graphs and equations of linear relationships?

y = x

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M2-1.4 Transformation of Lines

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Examples: Notes:

EQ: How can you use knowledge about geometric transformations to transform the graphs and equations of linear relationships?

Parallel Lines: • Lines that have the same slope

(6,7)(3,3)

(4,1)(5,3) Coincident: 2 lines or shapes that lie

exactly on top of each other.

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M2-1.4 Transformation of Lines

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Examples: Notes:

EQ: How can you use knowledge about geometric transformations to transform the graphs and equations of linear relationships?

Perpendicular Lines: • Lines that have slopes with the product of -1.• Negative reciprocals of each other.

(1,9)

(7,3)(1,4)

(2,0)

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