MATH 115
Graph Analysis
A graph analysis has three major components:



window analysis
sign analysis
direction analysis
Window Analysis
The window analysis is named after the WINDOW menu in your graphing calculator. A
window analysis sets the horizontal and vertical virtual window needed for viewing a
complete graph. A window analysis has two major components: domain and range.
Recall that a function is usually presented in one of four basic ways: table, equation,
graph, or reliable calculating machine. The domain




from a table consists of all entries in the input (usually first) column;
from an equation consists of all x values for which the equation computes a single
y-calue;
from a graph consists of all x-values for which there is a point on the graph;
from a machine consists of all inputs accepted by the machine.
While the domain of a function specifies all suitable inputs (x-values), the range of a
function specifies all suitable outputs. Hence, the range




from a table consists of all entries in the output (usually second) column;
from an equation consists of all y-values computed by the equation;
from a graph consists of all y-values for which there is a point on the graph;
from a machine consists of all obtainable outputs.
For example, consider the function displayed in Sullivan and Sullivan, Precalculus
Enhanced with Graphing Utilities, Fourth Edition, page 82, Figure 21.
y
(3,4)
(0,4)
(6,1)
(-6,0)
x
(-2,0)
(-4,-2)
1
The window analysis merely specifies the domain and the range of the given function.
Hence, for the above example, we report
domain: -6 ≤ x ≤ 6 or [-6,6] in interval notation
range: -2 ≤ y ≤ 4 or [-2,4] in interval notation
Sign Analysis
A basic observation is to note that a given point on the graph is either



below the x-axis
on the x-axis
above the x-axis.
When a point is below the x-axis, then f(x)<0, and vice-versa. When a point is on the xaxis, then f(x)=0, and vice-versa. When a point is above the x-axis, then f(x)>0, and viceversa.
A sign analysis summarizes these basic observations. A simple diagram that summarizes
the results portrays the x-axis, the x-intercepts, and an indication where the graph is
positive (above the x-axis) or negative (below the x-axis). For example,
y
(3,4)
(0,4)
(6,1)
(-6,0)
x
(-2,0)
(-4,-2)
[
-6
+

]
6 x
-2
Our geometric observations may now be summarized in a table:
sign
f(x) < 0
f(x) = 0
f(x) > 0
x
(-6,-2)
-6, -2
(-2,6]
Here, (-6,-2) denotes the open interval; i.e., -6 < x < -2.
Here, (-2,6] denotes the interval –2 < x ≤ 6.
2
Direction Analysis
The direction analysis describes for you, as you read the graph from left to right on the
horizontal axis, what is happening to the graph in the vertical direction. As you increase
x, does y increase? decrease? or remain constant? The formal definitions are as follows.
A function f is increasing on an open interval I means if x1 and x2 are numbers in the
interval I where x1 < x2, then f(x1) < f(x2). Colloquially, for x-values in the interval I, the
bigger the x-value the bigger the y-value. A function f is decreasing on an open interval I
means if x1 and x2 are numbers in the interval I where x1 < x2, then f(x1) > f(x2).
Colloquially, for x-values in the interval I, the bigger the x-value the smaller the y-value.
And, a function f is constant on an open interval I means if x1 and x2 are numbers in the
interval I where x1 < x2, then f(x1) = f(x2). Colloquially, for x-values in the interval I, the
y-value stays the same. A nontrivial interval is simply an interval that contains at least
two points; i.e., is neither the empty set nor a singleton set.
y
(3,4)
(0,4)
(6,1)
(-6,0)
x
(-2,0)
(-4,-2)
We now refer to the above example. We begin reading the graph at the leftmost point of
its domain: x = -6. Starting at x = -6, as we increase x, the y-values decrease from y = 0 at
x = -6 to y = -4 at x = -2. So, the function is decreasing on the open interval -6 < x < -2.
In interval notation, the function is decreasing on the open interval (-6,-2).
After x crosses -4, then the graph starts increasing until we reach x = 0. The function is
increasing on the open interval -4 < x < 0. In interval notation, the function is decreasing
on the open interval (-4,0).
After x = 0, the functional values stay constant at f(x) = 4 until x = 3. Hence, the function
is constant for 0 < x < 3. In interval notation, the function is constant on the open interval
(0,3). Finally, after x > 3, the function declines until it reaches x = 6. Hence, the function
is declining on the open interval (3,6).
We summarize these results in the following table.
We now present two examples that simply supply the
graph anaysis without the explanatory commentary.
3
direction
increasing
decreasing
constant
interval
(-4,0)
(-6,-4) and (3,6)
(0,3)
Example Given the squaring function, f(x) = x2.
y
Window Analysis:
domain: (-∞,∞) or -∞ < x < ∞
range: [0, ∞) or 0 ≤ y < ∞
Sign Analysis:
x
sign
f(x) = 0
f(x) > 0
x
x=0
(-∞,0) and (0, ∞)
Direction Analysis:
direction
x
decreasing (-∞,0)
increasing (0, ∞)
Example Given the function, y = f(x).
Window Analysis:
domain: (-∞,∞) or -∞ < x < ∞
range: (0, ∞) or 0 < y < ∞
y
Sign Analysis:
sign
f(x) > 0
x
(-∞,∞)
Direction Analysis:
x
4
direction
increasing
x
(-∞,∞)