Download Graphing Equations: An Ordered Pair of

Graphing Equations:
An Ordered Pair of numbers is two numbers (x, y) that is used to
represent coordinate of points in the Cartesian plane. The first number
is the x coordinate and the second number is the y coordinate. The
point ?(0, 0) is the origin and is at the intersection of the x and y axis.
First quadrant: x > 0, y > 0.
Second quadrant: x < 0, y > 0.
Third quadrant: x < 0, y < 0
Fourth quadrant: x > 0, y < 0.
Given an equation in two variables, x and y, the solutions to the equation can be represented as (x, y), where the first coordinate is the x
value and the second coordinate is the y value that makes the equation
true. When we do that, each solution (x, y) can now be viewed as a
point on the Cartesian plane. If we plot all the solutions in ordered
pair form (x, y) as points on the Cartesian plane, we get the graph of
the equation.
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E.g. Graph x + y = 1
One way to graph an equation in two variables, if we know nothing
about the equation, is to just plot points.
We pick different values of x and solve the equation for y:
For example, if x = 1, then y = 0, if x = 2, then y = −1, if x = 3,
then y = −2
x
0
1
-1
2
-2
3
y
1
0
2
-1
3
-2
If we plot all these ordered pair of points on the Cartesian plane, we
get the graph of a line. The above equation is a linear equation.
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Functions
Informal definition of a function:
A function between two sets is a rule that assigns to each member
in the first set (called the domain one and only one member in the
second set (called the range).
Intuitively, a function is a machine (or an operation) that takes an
input and produces an output based on the input. For example, think
of a vending machine that sells soft drinks. It takes money (in the
form of coins or bills) as input, and produces canned soda as output.
In algebra, the functions we are interested are functions that takes real
number as input and gives real numbers as output. An example of a
simple function would be something like:
3
-4
5
7
f:
→
→
→
→
3
2
1
-1
2
In order for an operation to be a valid function, each input must produces one and only one output. However, it is ok for a function to
produce the same output with different input. Notice in this example
here that f gives 2 as output for both the inputs 3 and 7, which is fine.
r:
11 →
6 →
11 →
-2 →
7
0
4
0
This is not a function because the input 11 has two output, 7 and 4.
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What makes a function useful is the fact that it expresses the relationship of two quantities, where one quantity (output) is determined by
the value of another quantity (input). For example, for a car driving at
a constant speed, if we know how many hours the car has been driven,
we know how much distance it has travelled. In this case, we say that
the distance is a function of time.
If we think of the input as the x coordinate and the output as the y
coordinate, then it is possible to describe a function using an equation:
E.g. y = 3x + 1
In this equation, if we treat y as the output and x as the input, then
since the value of y is completely determined by the value of x, this
implicitely describes a function. We say that y is a function of x.
E.g. y = x2 − 4
This also describes y as a function of x, since for each x there is one
and only one y that solves the equation.
E.g. |y| = 2x
For this equation y is not a function of x. For some values of x, there
are more than one y value that solves the equation. For example, if
x = 1, then y = 2 or y = −2 both solves the equation.
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It is important to know that while some equations (in two variable) describe a function, and some functions can be described by an equation,
an equation is not a function. A function is the operation (relationship)
behind the input (x) and output (y) that gives rise to the equation.
Function Notation
We use the f (x) (read f of x) notation to represent a function.
E.g.
f (x) = 3x − 1
Here, f is the name of the function, x is the input, and f (x) is the
output. This function notation is useful because it allows us to easily
express the values of f on each input:
f (1) = 3(1) − 1 = 3 − 1 = 2
f (−3) = 3(−3) − 1 = −9 − 1 = −10
f (a + 1) = 3(a + 1) − 1 = 3a + 3 − 1 = 3a + 2
A function may be graphed like an equation if it is understood that
the output is y and the input is x.
Vertical line test:
If any vertical line drawn intercepts a given graph at most once, then
the given graph is the graph of a function.
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Domain and Range
When a function is given as an equation in the f (x) notation, it is
understood that the domain of the function is all real numbers that
makes the function defined, and the range is all possible values of f
for x in the domain.
E.g. f (x) = x + 5
For this example, f is defined for any real number x, so domain of f is
all real numbers. Also evary real number can be produced for different
values of x, so the range of f is also all real numbers.
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E.g. f (x) =
x−1
For this example, if x = 1, then we have division by 0, and this will
make f undefined. For all other values f is defined, therefore, the
domain of f is all real numbers except 1.
The only number that f cannot produce is the number 0, therefore the
range of f is all real numbers except 0.
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Linear Functions
A linear function is a function of the form:
f (x) = mx + b
where m and b are constants.
The graph of f is a straight line, m is the slope of the line and b is
the y-intercept of the line.
The equation y = mx + b is called the slope-intercept form of the
equation of a line.
Given any two points (x1 , y1 ) and (x2 , y2 ) on a line, the slope of the
line can be found by:
y 2 − y1
m=
x2 − x1
E.g. Find the slope of the line containing the points (1, 4), (−2, 3)
3−4
−1 1
=
=
m=
−2 − 1 −3 3
1
The slope of the line is
3
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Geometrically, the slope of the line measures the slantedness of the
line. The larger the magnitude of the slope, the more slanted (close
to vertical) the line is; the smaller the magniture of the slope, the less
slanted (close to horizontal) the line is.
A vertical line has undefined slope.
A horizontal line has slope = 0.
Intuitively, the slope of a line measures the constant rate of change of
y for each unit change in x. For example, a slope of 2 means that,
whenever x increases by 1 unit, y increases by 2 units. A slope of −3
means that, whenever x increases by 1 unit, y decreases by 3 units.
E.g. If you are driving at a constant speed of 50 miles per hour. If
y represents the distance you travelled and x represents the time you
spent driving, then the 50mph is the slope (which is your speed).
The x-intercept of the graph is a point where y = 0 (it is where the
graph intercepts the x-axis.
The y-intercept of the graph is a point where x = 0 (it is where the
graph intercepts the y-axis.
To graph a line in slope intercept form, we may first locate the y
intercept on the graph, then move according to the slope.
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Finding Equations of Lines
Given enough information about a line (two points, or a point and the
slope), we can find the equation of a given line:
Point-Slope Form of the equation of a line:
y − y1 = m(x − x1 )
This form is nothing more than a restatement of the definition of the
slope. This form is most convenient when we have the slope m and a
point (x1 , y1 )
Standard Form
ax + by = c
This form is useful when the line in question does not have a slope (i.e
a vertical like) or if we need to solve systems of equations (later).
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Slope-Intercept Form
y = mx + b
This form gives the most geometrical information about the line, and
is the most convenient for graph. This is also the only unique form of
the equation of a line out of the three.
E.g.
Find the equation of the line with slope −3 and containing the point
(−2, 4)
We may put it in point-slope form since we already have a point and
a slope:
y − 4 = −3(x − (−2))
y − 4 = −3(x + 2)
y − 4 = −3x − 6
y = −3x − 2
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E.g.
Find the equation of the line containing the points (−3, 4) and (−2, −6)
We first find the slope of the line:
−6 − 4
−10
−10
m=
=
=
= −10
−2 − (−3) −2 + 3
1
With the slope, we may use the point-slope form again:
y − 4 = −10(x − (−3))
y − 4 = −10(x + 3)
y − 4 = −10x − 30
y = −10x − 26
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E.g. Find the equation of the vertical line containing the point (3, 5)
Answer: All points on a vertical line have the same x-coordinate, so
the equation is x = 3
E.g. Find the equation of the horizontal line containing the point (2, 7)
Answer: All points on a horizontal line have the same y-coordinate, so
the equation is y = 7
In general, any vertical line has an quation of the form x = a, where a
is the x-coordinate of any point on the line.
In general, any horizontal line has an quation of the form y = b, where
b is the y-coordinate of any point on the line.
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