Notesale: Turn your study into money

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Visualizing linear functions
The coordinate plane
As you remember from pre-algebra a coordinate plane is a two-dimensional number
line where the vertical line is called the y-axis and the horizontal is called the x-axis
...
This point is called
the origin
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A point in a coordinate plane is named by its ordered pair of the form of (x, y)
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The completeness property for points in the plane tells us two things
1
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Exactly one ordered pair of numbers names a given point in the plane
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It's always a good idea to start at the origin
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And the ycoordinate tells you have many steps to move up (positive) or down (negative)
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Begin at the point and follow a vertical line either up or down to the x-axis
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And then do the same but following a horizontal line to find the ycoordinate
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The first coordinate (usually the x-coordinate) is
called the domain and the second (usually the y-coordinate) is called the range
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Linear equations in the coordinate
plane
A linear equation is an equation with two variables whose graph is a line
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If all variables represent real numbers one can graph the equation by
plotting enough points to recognize a pattern and then connect the points to include
all points
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When choosing your points try to
include both positive and negative values as well as zero
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g
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X

Y=x+2

Ordered pair

-2

-2 + 2 = 0

(-2, 0)

-1

-1 + 2 = 1

(-1, 1)

0

0+2=2

(0, 2)

1

1+2=3

(1, 3)

2

2+2=4

(2, 4)

Now you can just plot the five ordered pairs in the coordinate plane

At the moment this is an example of a discrete function
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By drawing a line through all points and while extending the line in both directions we
get the opposite of a discrete function, a continuous function, which has an unbroken
graph
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The point in which the graph crosses the x-axis is

called the x-intercept and the point in which the graph crosses the y-axis is called the yintercept
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The standard form of a linear equation is

𝐴𝑥 + 𝐵𝑦 = 𝐶, 𝐴, 𝐵 ≠ 0
Before you can graph a linear equation in its standard form you first have to solve the
equation for y
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The graph of y = a is a horizontal line where the line passes through the point (0, a)

Whereas the graph of x = a is a vertical line that passes through the point (a, 0)

The slope of a linear function
The steepness of a hill is called a slope
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The
slope is defined a s the ratio of the vertical change between two points, the rise, to the
horizontal change between the same two points, the run
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(x1, y1) represents the first
point whereas (x2, y2) represents the second point
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Example
Find the slope of the line

(𝑥1, 𝑦1) = (−3, −2)𝑎𝑛𝑑 (𝑥2, 𝑦2) = (2,2)
𝑚=

𝑦2 − 𝑦1 2— (−2) 2 + 2 4
=
=
=
𝑥2 − 𝑥1 2 − (−3) 2 + 3 5

A line with a positive slope (m > 0), as the line above, rises from left to right whereas a
line with a negative slope (m < 0) falls from left to right
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In earlier chapters we have looked at how fast a car drives and talked about speed in
miles per hour
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The rate of change
compares a change in one quantity to a change in another quantity like at what
speed does a car travel if it travels 120 miles in 2 hours?
𝑣=

𝑑 120
=
= 60
𝑡
2

You can interpret a slope of a line as the rate of change
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Now we're going to focus on the slope-intercept
form y = mx + b
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𝑦 = 𝑚𝑥 + 𝑏
Where m is the slope and b is the y-intercept
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You can check to see that the line you've drawn is the correct one by substituting the
coordinates of the second point into the original equation
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Our second point = (1, 3)
𝑦 − 2𝑥 = 1

3−2x1= 3−2 = 1

Our second point is a solution to the equation i
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the line we drew is correct
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𝑦 = 𝑚𝑥

In a direct variation the nonzero number m is called the constant of variation
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You can use
other letters than f to name functions
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All functions that can be written on the form f(x) = mx + b belong to the family of
linear functions
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The
parent function of all linear functions is
𝑓(𝑥) = 𝑥



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