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Linear Relationships
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1. linear functions
- Proportionality
- Funzioni lineari - le basi
- Graph the Line
- Parameters of a linear equation
- Linear functions and growth rate
- EDSE 770 Activity - Linear functions
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2. parallel and perpendicular lines
- Parallel & Perpendicular Lines
- Exploration: Parallel and Perpendicular Lines
- Parallel & Perpendicular Consequence
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3. linear transformations
- Linear Function Transformations
- Mapping Diagrams and Graphs of Linear Functions
- linear functions - parameter plane
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4. linear regression
- Linear Regression
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5. linear inequalities
- Proportional Reasoning
- Solutions to Linear Inequalities
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Linear Relationships
Lee McCulloch-James, Sep 4, 2017
Table of Contents
- linear functions
- Proportionality
- Funzioni lineari - le basi
- Graph the Line
- Parameters of a linear equation
- Linear functions and growth rate
- EDSE 770 Activity - Linear functions
- parallel and perpendicular lines
- Parallel & Perpendicular Lines
- Exploration: Parallel and Perpendicular Lines
- Parallel & Perpendicular Consequence
- linear transformations
- Linear Function Transformations
- Mapping Diagrams and Graphs of Linear Functions
- linear functions - parameter plane
- linear regression
- Linear Regression
- linear inequalities
- Proportional Reasoning
- Solutions to Linear Inequalities
Proportionality
Parallel & Perpendicular Lines
PARALLEL LINES
The following applet demonstrates a property that parallel lines have when they're drawn in the coordinate plane.
Be sure to move the blue points around quite a bit!
QUESTION 1
If you move the blue points to make the lines vertical, are the lines consider parallel?
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No, since the two lines overlap they are coinciding. However, vertical line are still considered parallel when they are not coinciding.
What can you conclude about parallel lines drawn in the coordinate plane?
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They have the same slope yet different y-intercepts.
PERPENDICULAR LINES
This applet demonstrates a property that perpendicular lines have when they're drawn in the coordinate plane.
Be sure to move the points around quite a bit and observe carefully as you do!
What can you conclude about perpendicular lines that are drawn in the coordinate plane?
(Assume the lines are not aligned horizontally and vertically).
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Other than the case of horizontal and vertical lines, if two lines are drawn in the coordinate plane, then their slopes are always opposite reciprocals. Another way of expressing this fact is that their slopes always multiply to -1.
Linear Function Transformations
Linear Function Transformations
Linear Function Transformation Exercise
The linear function y = x, denoted by function g.
The slope-intercept form is y = mx + b, where m=slope and b=y-intercept of the function.
Note: The 'slider' feature on the x-y coordinate plane can be used to change the m and b values
for the following exercises. To do so, place the cursor and hold it on the dot of the slider and
slide it to the desired m and b values.
To move the slider to a different location on the x-y plane, place the cursor and hold it on the line
of the slider and move it to the desired location.
Note: You can zoom in or out with the mouse.
Exercise 1
Perform the following linear function transformation:
Vertical shift of 3 units up (y-intercept = 3).
The new function is y=x +3 , denoted by function f.
Set the slope of the function to m=1 by entering 1 for m.
Set the y-intercept of the function to b=3 by entering 3 for b.
Observe the transformation of the linear function.
Exercise 2
Perform the following linear function transformation:
Vertical shift of 3 units down (y-intercept = -3).
The new function is y=x - 3 , denoted by function f.
Set the slope of the function to m=1 by entering 1 for m.
Set the y-intercept of the function to b=-3 by entering -3 for b.
Observe the transformation of the linear function.
Exercise 3
Perform the following linear function transformation:
Change slope of the linear function to 2.
The new function is y=2x , denoted by function f.
Set the slope of the function to 2 by entering 2 for m.
Set the y-intercept of the function to b=0 by entering 0 for b.
Observe the transformation of the linear function.
Exercise 4
Perform the following linear function transformation:
Change slope of the linear function to - 2.
The new function is y= - 2x ,denoted by function f.
Set the slope of the function to -2 by entering -2 for m.
Set the y-intercept of the function to b=0 by entering 0 for b.
Observe the transformation of the linear function.
Exercise 5
Perform the following linear function transformation:
Graph a constant linear function by changing the slope of the
linear function to 0 with a y-intercept of 3.
The new function is y= 0 +3 = 3 , denoted by function f.
Set the slope of the function to 0 by entering zero for m.
Set the y-intercept of the function to 3 by entering 3 for b.
Observe the transformation of the linear function.
Exercise 6
Repeat this exercise as many times as desired until concept is mastered.
Use different values of m and b.
Linear Regression
Linear Regression
-Download Geogebra File: Linear Regression
-Move some of your points around so that your value of r goes from -1 to 1
-Move some of your points around so that the equation of Regression has a gradient of 2 and a y-intercept of 3
-What happens to your value of r?
Proportional Reasoning
Move the point E up or down to change the size of the man on the right.
Proportional Reasoning
Does the ratio of the size of the man to the size of the line look the same as the man and line on the left?
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