MATH 115: Algebra and Elementary Functions

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) (-2,0) x

(-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 x- axis, then f(x)=0, and vice-versa. When a point is above the x-axis, then f(x)>0, and vice- versa.

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) (-2,0) x

(-4,-2)

+ [ ]  -6 -2 6 x

Our geometric observations may now be summarized in a table:

sign x f(x) < 0 (-6,-2) Here, (-6,-2) denotes the open interval; i.e., -6 < f(x) = 0 -6, -2 x < -2. f(x) > 0 (-2,6] 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) (-2,0) x

(-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. direction interval increasing (-4,0) decreasing (-6,-4) and (3,6) constant (0,3)

3 We now present two examples that simply supply the graph anaysis without the explanatory commentary.

Example Given the squaring function, f(x) = x2.

y Window Analysis: domain: (-∞,∞) or -∞ < x < ∞ range: [0, ∞) or 0 ≤ y < ∞

Sign Analysis:

sign x x f(x) = 0 x = 0 f(x) > 0 (-∞,0) and (0, ∞)

Direction Analysis:

direction x decreasing (-∞,0) increasing (0, ∞)

Example Given the function, y = f(x).

y Window Analysis: domain: (-∞,∞) or -∞ < x < ∞ range: (0, ∞) or 0 < y < ∞

Sign Analysis:

sign x f(x) > 0 (-∞,∞)

Direction Analysis:

direction x increasing (-∞,∞) x