Chapter 3-1 Graphing
Linear Equations
There are FOUR types
of linear graphs and this chapter begins with an OVERALL,
BIG picture
Positive Slope-
slants up from left to right
Negative Slope- slants down from left to right
Horizontal line- stays flat from left to right ( constant function)
Vertical Line- stays straight up and down ( Not a function—why??)
Negative Slope- slants down from left to right
Horizontal line- stays flat from left to right ( constant function)
Vertical Line- stays straight up and down ( Not a function—why??)
Somethings to look
for:
Domain
Range
End behavior
Intercepts
Extrema
Positive/Negative
Increasing/Decreasing
Symmetry
Domain
Range
End behavior
Intercepts
Extrema
Positive/Negative
Increasing/Decreasing
Symmetry
A Linear Equation
is an equation that forms a line when it is graphed. Linear equations are often
written in the form Ax + By = C
This is called standard form. In this equation C is called a constant Ax and By are variable terms.
This is called standard form. In this equation C is called a constant Ax and By are variable terms.
A ≥ 0
A and B BOTH cannot be 0
A, B,and C are ALL integers with a GCF= 1
A and B BOTH cannot be 0
A, B,and C are ALL integers with a GCF= 1
If you see a term
such as xy attached to together it cannot be a linear equation. If the exponent
on a variable is different than the understood 1, it is not a linear equation
in 3x + 2y = 5
A = 3
B = 2
C = 5
B = 2
C = 5
In x = -7 ( Yes that
is in Standard Form)
A = 1
B = 0
C = -7
B = 0
C = -7
Identify Linear
Equations
Determine whether
each equation is a linear equation.
Write the equation in Standard form
y = 4 – 3x YES
y = 4 – 3x YES
To put this
equation in standard form, we need to move the -3x term to the other side,
using the Addition Property of Equality
and the Additive Inverse Property. So
that the x and y values are on the SAME side and the constant is always on the
other side to the right of the equal sign.
3x + y = 4
A = 3
B = 1
C = 4
3x + y = 4
A = 3
B = 1
C = 4
6x –xy = 4 NO
the term xy has
two variables the equation cannot be written in AX + By = C . It is not a
linear equation
(1/3)y = -1 Yes
It becomes y = -3
A= 0
B = 1
C = -3
B = 1
C = -3
A linear equation
can be represented on a coordinate graph. The x- coordinate of the point at which
the graph of the equation crosses the x-axis is called the x-intercept. The y- coordinate of the point at which the graph of the
equation crosses the y-axis is called the y-intercept.
The graph of linear equation has AT MOST one x- intercept
and ONE y-intercept ( unless it is the equation x = 0, which is the y-axis or y
= 0, which is the x-axis. In those two special cases every number is a
y-intercept or an x-intercept, respectively)
Real World Example
Swimming Pool Page 157 in your textbook
A swimming pool is
being drained at a rate of 720 gallons per hour. The table on Page 157 shows
the function relating the volume of water in a pool and the time in hours that
the pool has been draining.
Find the x- and y-
intercepts on the graph of the function.
Looking at the
table we see that the x intercept is 14 ( that is when y is 0)
and the y-intercept is 10,080 ( that is the value of y, when x = 0)
and the y-intercept is 10,080 ( that is the value of y, when x = 0)
Describe what the
intercepts mean in this situation: This should remind you of our unit at the
beginning of the year!
The x intercept 14
means that after 14 hours the pool is completed drained because it has a volume
of 0 gallons!
The y- intercept
of 10,080 means that the pool contained 10,080 gallons of water at time 0 ( or
before it started to drain)
Graph by Using Intercepts
Graph 2x + 4y = 16 using just the x-intercept and y-intercept
Graph 2x + 4y = 16 using just the x-intercept and y-intercept
2x + 4(0) =
16 replace y with 0 (or as taught in
class cover over the y value and solve)
2x = 16 so x = 8 (
when y = 0) ( 8,0)
This means the
graph intersects the x-axis at (8,0)
Now
2(0) + 4y =
16 replace x with 0 ( or as taught in
class- cover over the x value and solve)
4y = 16
y = 4 ( when x = 0) ( 0, 4)
y = 4 ( when x = 0) ( 0, 4)
This means the
graph intersect the x-axis at (0, 4)
Plot these two
point and draw a line through them
Notice that this
has both an x- intercept and y-intercept
Some lines have
only an x- intercept and NO y-intercept
or vice versa
y = b is a
horizontal line that has only a y- intercept (unless b=0)
The graph of x = a
is a vertical line that has only an x- intercept (unless a = 0)
Lines that are neither
vertical or horizontal cannot have more than one x- and/or y-intercept.
Graphing Using an XY Table
Another way to
graph is choosing random x values , plugging those into the equation to find
the corresponding y values, and graphing those points you found.
Although 2 points
determine a line, it is always best to find 3 points so that you are sure you
did not make a mistake on either of the first two points.
If the coefficient
of x is a fraction, select a value that is
multiple of the denominator so hopefully you won’t end up with fractions
to graph!
