11.05 Characteristics from a graph

Intervals of increasing, decreasing and constant

One of the key characteristics of a graph are the intervals of increase, decrease and constant slope or rate of change. The intervals will always be in terms of the independent variable, usually $x$x. We can write these intervals in inequality or interval notation.

On the graph below, we can see what the different scenarios look like.

• If as values of $x$x increase, the values of $f(x)$f(x) increase, then the function is increasing over that interval
• If as the values of $x$x increase, the values of $f(x)$f(x) decrease, then the function is decreasing over that interval
• If as the values of $x$x increase, the values of $f(x)$f(x) remain constant, then we can say the function is constant over that interval

We can also interpret this in terms of the rate of change of the variables. For example, acceleration is the rate of change of velocity with respect to time. If the acceleration is positive (so that the slope of the velocity with respect to time is positive) then the velocity is increasing, and if the acceleration is negative then the velocity is decreasing.

 

Worked example

Question 1

State the intervals of increase and decrease for the graph below.

Think: This is a parabola. To the left of the vertex the function is decreasing and to the right of the vertex the function is increasing. At the vertex, it is switching from decreasing to increasing, so is technically constant at that exact point.

Do: Let's write our answer in both inequality and interval notation. They will be in terms of the $x$x-values.

Interval of decrease: $x<0$x<0 which can be written as $x\in(-\infty,0)$x∈(−∞,0)

Interval of increase: $x>0$x>0 which can be written as $x\in(0,\infty)$x∈(0,∞)

Reflect: At $x=0$x=0 the function is neither increasing nor decreasing, so we use an open interval to exclude $0$0.

 

Question 2

A company's daily revenue, $R$R, in thousand dollars over $12$12 months has been plotted below. We want to classify the entire domain of the function graphed below into regions which are increasing, or decreasing:

Think: While we do not know exactly what this function is, we have enough information in the graph to work this out. We will start by finding the points where it changes from increasing to decreasing or vice versa.

Do: First, notice points where the graph changes from increasing to decreasing or vice versa at $t=3$t=3 and $t=5$t=5. At $t=10$t=10, it flattens out, but does not actually change to decreasing, we will look at these kinds of points more in future studies. 

Intervals of increase: $00<t<3 and $55<t<12 which can also be written as $t\in\left(0,3\right)\cup\left(5,12\right)$t∈(0,3)∪(5,12)

Intervals of decrease: $33<t<5 which can also be written as $t\in\left(3,5\right)$t∈(3,5).

Reflect: Knowing the intervals when the function is increasing and decreasing lets us interpret the rate of change of the revenue. Since the function is always positive, the company is always taking in money.

This is true even when the function is decreasing. The rate of change is negative on the interval $\left(3,5\right)$(3,5), so the company is taking in less money during this period than they did at the start of month $3$3 but the rate of change is slow enough that the revenue never completely stops.

 

Practice questions

Question 3

 

Intercepts, extrema and end behavior

We're now going to identify some key characteristics of non-linear graphs without being given their equations.

We'll be looking for the following features of the graphs:

$x$x-intercepts	Points where the graph crosses the $x$x-axis and $f(x)=0$f(x)=0.
$y$y-intercept	The point where the graph crosses the $y$y-axis and $x=0$x=0. If the graph is a function, there will be only one of these.
Absolute Maxima/Minima	The greatest/least value $y$y takes across the whole domain. This appears on the graph as a turning point. Sometimes, when the graph has no upper/lower limit, this will not exist.
Local Maxima/Minima	The greatest/least value $y$y takes in a particular region. This appears on the graph as a turning point.
Slope	The steepness of the graph. The slope is zero when flat, more positive for 'uphill' slopes, and more negative for 'downhill' slopes.
Extrema Behavior	Does the graph increase or decrease as $x\to\infty$x→∞ or $x\to-\infty$x→−∞?

Questions might also ask about key features of a graph, but only in a particular region, by specifying a domain.

 

Worked example

Question 4

Consider the following graph.

a) What is the $y$y-intercept of the graph?

b) How many $x$x-intercepts does the graph have? What are they?

c) How many local maxima does the graph have? How many local minima does it have?

d) Is the graph increasing or decreasing as $x\to\infty$x→∞? How about as $x\to-\infty$x→−∞?

e) Does the graph have an absolute maximum? Does it have an absolute minimum?

f) In which domain will you find the greatest positive slope?

(A) $\left(-4,-2\right)$(−4,−2)

(B) $\left(-2,1\right)$(−2,1)

(C) $\left(1,2\right)$(1,2)

(D) $\left(2,6\right)$(2,6)

SOLUTION

a) The graph crosses the $y$y-axis here.

Hence, the $y$y-intercept is $y=-2$y=−2.

b) The graph has five $x$x-intercepts at the following points. They are $x=-4$x=−4, $x=-2$x=−2, $x=1$x=1, $x=2$x=2 and $x=6$x=6.

Notice that $x=6$x=6 is still an intercept even though it only touches the axis rather than crossing through it.

c) The graph has three local maxima at the following points. They are the maximum values in the local region around them.

The graph has two local minima at the following points. 

d) As $x\to\infty$x→∞ and $x\to-\infty$x→−∞ the graph is decreasing on both sides towards negative infinity.

e) The graph has an absolute maximum at the following point, since the graph never goes higher than this.

However, even though we have two local minima, there is no absolute minimum because the graph decreases without bound on either side as $x\to\infty$x→∞ and $x\to-\infty$x→−∞, so never finds a minimum.

f) The domains being referred to are these regions of the graph between the $x$x-intercepts (the hills and valleys).

The solid lines below show the parts of the graph in these regions that only have positive slopes.

We can see from this that the steepest positive slope occurs in the interval $\left(-4,-2\right)$(−4,−2). Hence, (A) is the correct answer.

 

Practice questions

Question 5

Question 6

Question 7

 

Asymptotes

An asymptote of a function is a straight line which the function values approach under certain conditions. There are three types of asymptotes: horizontal, vertical and oblique.

Horizontal asymptotes have equations of the form $y=c$y=c. They occur when the function approaches a constant value $c$c as $x$x tends towards positive or negative infinity.

Vertical asymptotes have equations of the form $x=c$x=c. They occur when the function values tend towards positive or negative infinity as $x$x approaches the constant value $c$c.

It is also possible for a function to have an oblique asymptote or slanted asymptote. Graphically, this means that the function approaches a straight line with a non-zero slope as $x$x tends towards positive or negative infinity.

As an example, here is the graph of a rational function:

Notice that the function values tend towards $\pm\infty$±∞ as the values of $x$x approach $1$1. This function has a vertical asymptote at $x=1$x=1, which has been displayed as a dashed line.

There is also a horizontal dashed line with equation $y=2$y=2, which is the horizontal asymptote of this function. We can see that the function values approach $2$2 as the values of $x$x tend towards positive and negative infinity.

 

 

Practice questions

Question 8

Question 9