Topics
5. Functions & Linear Relationships
topic badge
United States of AmericaTennessee
Integrated Math 1

5.04 Characteristics of functions

Lesson
Practice
Lesson

Concept summary

The important characteristics, or key features, of a function or relation include its

  • domain
  • range
  • x-intercepts
  • y-intercepts
  • maximum value (the highest output value)
  • minimum value (the lowest output value)
  • rate of change over specific intervals
  • end behavior (the value of y as x\to\pm\infty)

Key features of a function are useful in helping to sketch the function, as well as to interpret information about the function in a given context. Note that not every function will have each type of key feature.

The rate of change of a function over a specific interval can be broadly categorized by one of the following three descriptions:

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y

A function is increasing over an interval if, as the input values become higher, the output values also become higher.

A function is decreasing over an interval if, as the input values become higher, the output values become lower.

A function is constant over an interval if, as the input values become higher, the output values remain the same.

Worked examples

Example 1

Consider the function shown in the following graph:

-2
-1
1
2
3
4
5
6
7
8
x
-6
-4
-2
2
4
6
8
10
12
14
y
a

Identify whether the function has a maximum or minimum value, and state this value.

Solution

This function has a minimum value of -4.

b

State the range of the function.

Approach

In part (a) we identified that the function has a minimum value of -4. So we know that the function can't take values smaller than -4.

Solution

Looking at the function, we can see that it stretches up towards infinity on both sides of the minimum point. So the function can take any value greater than or equal to -4. That is, the range of the function is \text{Range: } \left\{y\, \vert\, y \geq -4\right\}

c

State the x-intercept(s) of the function.

Approach

The x-intercept(s) of a function are the points where the function crosses the x-axis. In this case, by looking at the graph we can see that there are two x-intercepts.

Solution

The x-intercepts of this function are the points \left(1, 0\right) and \left(5, 0\right).

d

Determine the largest interval over which the function is increasing.

Approach

In part (a) we identified that the function has a minimum value. Looking at the graph, we can see that the function values are decreasing on the left side of the minimum and increasing on the right side.

Solution

The minimum point occurs at x = 3. The function values are increasing for all x-values to the right of this minimum. That is to say, the function is increasing for x > 3.

Example 2

Sketch a graph that has the following features:

  • Domain of -7<x\leq2
  • Range of -6\leq x<6
  • Increasing on -1<x<2

Approach

When sketching a graph with specific features, we want to start with the given information, then fill in the rest of the sketch to connect the given parts.

For this graph, we know that the lower end of the domain, x=-7, and the upper end of the range, y=6, are not included. Since the domain has no gaps, this tells us that we want an open point at \left(-7,6\right).

Since the graph will be increasing on -1<x<2, we also know that the point at x=2 will have a y-value greater than -6.

Solution

Here is an example of a graph that has all the required features.

-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
x
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
y

Reflection

There are many different graphs that will have the required properties, but it is useful to make the graph as simple as possible to keep track of all its features.

Here is an example of a graph that has all the required features but is very complicated:

-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
x
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
y

Example 3

A penguin is tagged with a tracker to record its height above sea level when hunting. The height of the penguin is graphed against time.

5
10
15
20
25
t\left(\text{mins}\right)
-50
-40
-30
-20
-10
10
20
30
40
50
h\left(\text{ft}\right)

Use the key features of the graph to describe the penguin's time spent hunting. Be as detailed as possible.

Approach

We can see that the graph has key features like intercepts, a minimum point, increasing and decreasing regions, and a domain.

To interpret the graph into context, we can use the axes of the graph to match key features to their real-world meaning.

Solution

The y-intercept is \left(0,30\right), meaning that the penguin is 30\text{ ft} above sea level at 0 minutes into its hunting time.

The x-intercepts are \left(5,0\right) and \left(20,0\right), meaning that the penguin is exactly at sea level at 5 and 20 minutes into it's hunting time.

The minimum of the graph is approximately \left(12.5,-41\right), so the penguin's lowest point is about 41\text{ ft} below sea level at about 12.5 minutes into it's hunting time.

If we combine this information with the increasing and decreasing regions of the graph, we can make a description of the penguin's hunting time. For example:

When the penguin needs to hunt, it leaves its nest which is 30 \text{ ft} above sea level. The penguin makes its way down to the water and dives into the water at 5 minutes after leaving the nest. The penguin swims down to a depth of around 41\text{ ft} below sea level, reaching its deepest point around 12.5 minutes into its hunting time, before returning to the water's surface at 20 minutes. The penguin spends 15 minutes underwater in total. The penguin then spends the last 5 minutes of its hunting time climbing back up to its nest, finishing a bit higher than where it started.

Reflection

We only need to make sure that the description matches the key features of the graph, so there are many possible examples.

For example, it is completely valid to say that the penguin dives into the water using a submarine as long as the deepest point is still 41\text{ ft} below sea level.

Outcomes

M1.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.*

M1.MP1

Make sense of problems and persevere in solving them.

M1.MP2

Reason abstractly and quantitatively.

M1.MP3

Construct viable arguments and critique the reasoning of others.

M1.MP4

Model with mathematics.

M1.MP5

Use appropriate tools strategically.

M1.MP6

Attend to precision.

M1.MP7

Look for and make use of structure.

What is Mathspace

About Mathspace