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2. Functions & Relations
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Algebra 1

2.04 Characteristics of functions

Lesson
Practice

Characteristics of functions

The important characteristics, or key features, of a function or relation include the previously seen domain, range, and the following additional features:

A parabola plotted in a four quadrant coordinate plane. The parabola opens downward, and passes through points (1, negative 4), (0, negative 3), (negative 1, 0), and (3, 0). The point (1, negative 4) is labeled minimum, (0, negative 3) labeled y-intercept, and (negative 1, 0) and (3, 0) labeled x-intercepts.

  • x-intercept (s) : (also called zeros) the point(s) where a graph intersects the x-axis. A function can have multiple x-intercepts.

  • y-intercept : the point where a line or graph intersects the y-axis. A function can only have up to one y-intercept.

  • minimum: the lowest output of a function

  • maximum: the highest output of a function

These key features apply to a variety of functions, as we see from the examples below:

-4
-3
-2
-1
1
2
3
4
-4
-3
-2
-1
1
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3
4

  • Domain: \left\{x \middle\vert x \in \Reals \right\}

  • Range: \left\{y \middle\vert y \leq 4 \right\}

  • x-intercepts: \left(-1,0\right),\left(3,0\right)

  • Zeros: \left(-1,0\right),\left(3,0\right)

  • y-intercept: \left(0,3\right)

  • Maximum: \left(1,4\right)

-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
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8
9

  • Domain: \left\{x \middle\vert -3 \leq x \leq 4 \right\}

  • Range: \left\{y \middle\vert -6 \leq y \leq 8 \right\}

  • x-intercept: \left(1,0\right)

  • Zeros: \left(1,0\right)

  • y-intercept: \left(0,2\right)

  • Maximum: \left(-3,8\right)

  • Minimum: \left(4,-6\right)

Examples

Example 1

Consider the function shown in the graph:

-2
-1
1
2
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4
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x
-6
-4
-2
2
4
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14
y
a

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

Worked Solution
Create a strategy

We need to find the lowest point on the graph and use the y-value to indicate how low it is.

Apply the idea
-2
-1
1
2
3
4
5
6
7
8
x
-6
-4
-2
2
4
6
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14
y

This function has a minimum value of -4.

b

State the range of the function.

Worked Solution
Create a strategy

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.

Apply the idea

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.

Worked Solution
Create a strategy

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.

Apply the idea
-2
-1
1
2
3
4
5
6
7
8
x
-6
-4
-2
2
4
6
8
10
12
14
y

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

Example 2

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.

Worked Solution
Create a strategy

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

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

Apply the idea

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 its hunting time.

If we combine this information, 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 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.

Reflect and check

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.

Example 3

A hiker's elevation over a given period of time is graphed:

A first quadrant coordinate plane with x-axis labeled as Time and y-axis labeled as Elevation. A piecewise graph is plotted. Speak to your teacher for more information.
a

What could the zeros of the function represent in this situation?

Worked Solution
Create a strategy

The zeros of a function represent where f\left(x\right)=0.

Apply the idea

The y-axis represents elevation and the x-axis represents time.

The zeros of this function represent the times where the hiker reached ground level.

b

Write a description of the meaning of the maximum of the function in relation to the hiker's elevation over time.

Worked Solution
Create a strategy

The maximum is the highest point of the function.

Apply the idea

The function relates the elevation of the hiker over a period of time.

The maximum represents the time where the hiker reached their highest elevation on their journey.

c

Would the domain or range tell us how long the hiker traveled?

Worked Solution
Create a strategy

The domain represents the inputs or x-values of a function, while range represents the y-values.

Apply the idea

The x-axis is labeled as time.

The domain represents how long the hiker traveled.

Idea summary

The key features of a function and how to describe them are as follows:

  • We can write intercepts, zeros, minimums, and maximums as values or ordered pairs.

  • Domain and range: write in inequality, set notation, and interval notation.

Outcomes

A.F.1

The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships.

A.F.1a

Determine and identify the domain, range, zeros, slope, and intercepts of a linear function, presented algebraically or graphically, including the interpretation of these characteristics in contextual situations.

A.F.2

The student will investigate, analyze, and compare characteristics of functions, including quadratic and exponential functions, and model quadratic and exponential relationships.

A.F.2a

Determine whether a relation, represented by a set of ordered pairs, a table, a mapping, or a graph is a function; for relations that are functions, determine the domain and range.

A.F.2b

Given an equation or graph, determine key characteristics of a quadratic function including x-intercepts (zeros), y-intercept, vertex (maximum or minimum), and domain and range (including when restricted by context); interpret key characteristics as related to contextual situations, where applicable.

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