11.05 Characteristics from a graph

Lesson

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)

True

A

False

B

True

A

False

B

Outcomes

F.IF.B.4'''

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. '''Include rational, square root and cube root; emphasize selection of appropriate models.