Topics
2. Linear functions
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United States of America
Grades 9-12

2.04 Characteristics of functions

Lesson
Practice

Introduction

Functions can be described by their key features. Some key features are intervals that describe how the function behaves, and other features like the intercepts, are individual points that carry a specific meaning. The key features of a function help us interpret and understand functions that describe real-world phenomena.

Characteristics of functions

The important characteristics, or key features, of a function or relation, include domain, range, and average rate of change over specific intervals, along with the following:

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): 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

We can use intervals to describe the behavior of the function, including where the function is increasing, decreasing, or constant, its domain and range, and where the function is positive or negative. When identifying intervals, except in the case of the range, we use the x-values to mark the beginning and end of the behavior.

A piecewise function composed of three lines plotted in a four quadrant coordinate plane. A blue line passing through points (negative 5, negative 4) and (negative 2, 3) labeled as Increasing, a green line passing through points (negative 2, 3) and (2, 3) labeled as Constant, and a purple line passing through points (2, 3) and (5, negative 2) labeled as Decreasing.
Increasing interval (of a function)

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

Decreasing interval (of a function)

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

Constant interval (of a function)

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

We can identify some of the key features from a graph using ordered pairs, inequality notation, interval notation, or set-builder notation, as shown:

A line plotted in a four quadrant coordinate plane, passing through point (1.5, 0) and is leaning to the left. The interval between the end of negative x-axis and x at 1.5 is labeled positive interval and the interval between the end of positive x-axis and x at 1.5 is labeled negative interval. The end of the line that lies in the second quadrant is labeled End behavior as x goes to negative infinity.The end of the line that lies in the fourth quadrant is labeled End behavior as x goes to infinity.

  • Domain: \left\{ x | -\infty < x <\infty \right\}

  • Range: \left\{y | -\infty < y <\infty \right\}

  • x-intercept: (1.5, 0)

  • Average rate of change over 0 \leq x \leq 3: -\dfrac{2}{3}

  • End behavior: As x \to \infty, the values of y become very large in the negative direction. As x \to -\infty, the values of y become very large in the positive direction.

  • Positive interval: (-\infty, 1.5]

  • Negative interval: (1.5, \infty)

  • Interval of decrease: -\infty < x <\infty

End behavior

Describes the trend of a function or graph at its left and right ends; specifically the y-value that each end obtains or approaches

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.

Examples

Example 1

Consider the function shown in the following graph:

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-6
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a

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

Worked Solution
Apply the idea

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

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.

Worked Solution
Create a strategy

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.

Apply the idea

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
Worked Solution
Create a strategy

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.

Apply the idea

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

-8
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x
-7
-6
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y
Reflect and check

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
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-5
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x
-7
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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
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t\left(\text{mins}\right)
-50
-40
-30
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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, increasing and decreasing regions, and a domain.

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 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 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 4

Use key features to sketch a graph of the following scenario:

A hiker is hiking up a mountain, and their initial elevation is at 0 feet above sea level. As the hiker begins on the trail, they have enough energy to climb their way to the first summit at an increasing pace, but they slow down and eventually take a lunch break on a ridge. The hiker continues their walk up the mountain at a steady rate after lunch and takes a short break at the summit, where they've reached the maximum elevation for the day. The hiker makes their way back down the mountain at a steady pace.

Worked Solution
Create a strategy

Key features throughout the story give us information about the shape of the graph, such as climbing at an increasing pace or when the hiker takes a break and climbs at a steady rate. Based on the story, the independent variable is the time, and the dependent variable is the elevation of the hiker.

Apply the idea

The hiker starts at 0 feet above sea level, so the y-intercept is at (0, 0), which is also the first x-intercept.

The description of the average rate of change along the first part of the hike is "at an increasing pace, but they slow down," so the graph increases at an increasing rate until it is constant for the lunch break.

There is an increasing interval at a steady rate to the maximum elevation the hiker gets to during the hike, and then the decreasing interval back down the mountain is at a steady rate until they reach the bottom, where the other x-intercept is located at (t, 0).

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.
Idea summary

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

  • Average rate of change: state as a number
  • Intercepts, minimums, and maximums: write as ordered pairs
  • Domain, range, positive and negative intervals, increasing and decreasing intervals: write in inequality, interval, or set-builder notation
  • End behavior: write in the form As x \to -\infty, f\left(x\right) \to ... and As x \to \infty, f\left(x\right) \to ...

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.

F.IF.C.7

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

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