Topics
4. Functions & Linear Relationships
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United States of AmericaFL
Grade 912

4.04 Characteristics of functions from a graph

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

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.

Outcomes

MA.912.AR.3.7

Given a table, equation or written description of a quadratic function, graph that function, and determine and interpret its key features.

MA.912.AR.5.6

Given a table, equation or written description of an exponential function, graph that function and determine its key features.

MA.912.F.1.5

Compare key features of linear functions each represented algebraically, graphically, in tables or written descriptions.

MA.912.F.1.6

Compare key features of linear and nonlinear functions each represented algebraically, graphically, in tables or written descriptions.

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