1.3 Graphs of Functions

Intro

If you've ever ridden a roller coaster you know how it feels to crawl up a hill and rocket down the other side. You also know how your stomach floats at the top and sinks at the bottom. It would be a very boring ride if the track never changed direction. The twists and turns, peaks and valleys make a roller coaster exciting and give it character.

The same can be said for functions: they wouldn't be interesting (or very useful) if they didn't change.

In this section you will learn to recognize features of a function by looking at a graph, in much the same way that you could pick out the loops and dips of a roller coaster from a photograph.

Graphing Functions with Technology

When we first encounter a new function we will generally want to see its graph. The equation will also be an essential tool, but lots of information can be gleaned from a quick look at the graph.

In the previous section we showed how the equation of a function can be used to create a table of values that can be plotted to create a graph. While knowing how to do that by hand is a useful skill, the technology available today makes exploring graphs of functions much simpler.

Between a graphing calculator, apps on your phone, and dedicated websites like Desmos.com or Geogebra.org, we will assume that you able to use technology to draw a graph of a function.

This section will help you read the graph of function and identify the features you see.

Increasing, Decreasing and Constant

When we read the shape of a function's graph, we always read from left to right, just like we read words in a sentence. For instance, a function increases if its graph goes up as we move from left to right and decreases if it goes down. These behaviors are also referred to as growth and decay, respectively. If the graph is flat then the function is constant on that interval; it does not grow or decay.

Consider the following graph.

This function is increasing on the interval $(a,b)$, constant on $(b,c)$, decreasing over the interval $(c,d)$ and increasing again on $(d,e)$.

Maximums and Minimums

Any point where a function changes from increasing to decreasing is known as a local maximum. On a graph it looks something like this.

If $f$ has a local maximum at $x=a$, then $f(a)$ is greater than any other function value in its immediate neighborhood, to the left or right.

If it turns out that $f(a)$ is the largest value of the function over its entire domain, then $f(a)$ is called a global maximum

Likewise, the point where a function changes from decreasing to increasing is called a local minimum.

If $f$ has a local minimum at $x=b$, then $f(b)$ is less than any other function values in the immediate neighborhood.

If it happens that $f(b)$ is the lowest value over the entire domain of the function, then $f(b)$ is called a global minimum.

Maximum and minimum points are important in many real life applications. For instance, we might want to minimize cost, or maximize fuel efficiency, or minimize waste.

The process of finding exact maximum and minimum values is called optimization and is properly studied in calculus. For our needs, however, a visual inspection of the graph will usually be accurate enough. When more precision is needed, the maximum and minimum commands found on most graphing utilities can be used.

Intercepts

In addition to maximum and minimum points, we should also look for points where the graph intersects the axes. The point where the graph crosses the y-axis is called the y-intercept. An x-intercept is a point where the graph intersects the x-axis.

Both of these concepts should be familiar to you from earlier algebra courses. The only new thing to be aware of is that x-intercepts will sometimes be called real zeros. This is because those x-values are real numbers that make the function equal to zero.

Concavity

Another important feature of a graph is its curvature, also known as its concavity. If a graph bends up, as if to form the side of a cup, then we say it is concave up on that interval. If the graph bends down, like a frown, then it is concave down. If the graph is straight then it does not bend and does not have any concavity.

Note that concavity does not tell us if a graph is increasing or decreasing. Instead, it gives us information about how quickly, or slowly, that change is happening.

For instance, a function that is both increasing and concave down will see its rate of growth slow down. But the growth rate of an increasing, concave up function will speed up. Rates of change will be studied in detail in the next section.

Asymptotes

Sometimes a function increases (or decreases) without bound toward infinity as it approaches a particular $x$ value. When this happens we say the function has a vertical asymptote at that $x$ value. In this example there is a vertical asymptote at $x=-1$.

Formally, the vertical line $x=c$ is a vertical asymptote of a function if $f(x)$ approaches either positive or negative infinity as $x$ gets near to $c$,. This is sometimes written $f(x)\rightarrow\pm \infty$ as $x \rightarrow c$.

We will use dotted vertical lines to indicate vertical asymptotes. Very few graphing utilities will draw asymptotes. You will have to infer their locations from the shape of the graph.

It is also possible for a function to eventually level off to the left or to the right. When this happens we say the function has a horizontal asymptote. In the example below, the function has a horizontal asymptote at $y=3$.

Technically speaking, the horizontal line $y=k$ is a horizontal asymptote of a function if $f(x)\rightarrow k$ as $x \rightarrow \pm \infty$.

Horizontal asymptotes describe a type of end behavior of a function. The end behavior of a function is how the function changes as $x$ approaches positive infinity or negative infinity. When looking for end behaviors, it doesn't matter how the function behaves in any local neighborhood, it only matters what happens far out at the ends. We will see functions with various combinations of increasing, decreasing and leveling off end behaviors.

Since a function can only have two end behaviors (one as $x \rightarrow \infty$ and another as $x \rightarrow -\infty$), a function can only have, at most, two horizontal asymptotes. There is no such limit on the number of vertical asymptotes.

Complete Graphs

A graph that indicates all the important features of a function is called a complete graph. If a function has any intercepts, asymptotes, maximums and/or minimums, then those should be visible on a complete graph.

Creating a complete graph can be a difficult adventure. Very few, if any, graphing programs will automatically give you a complete graph. However, they all have commands for zooming in and out and changing the viewing window.

As you learn more about different types of functions, you will be able to reduce the amount of guess work by recognizing how elements of the equation affect the shape of the graph. For now, however, you will want to experiment with the different zoom and window commands available.

Determine Domain and Range from a Graph

A complete graph can help us identify the domain and range of a function. Since each point on the graph has an $x$ and a $y=f(x)$ coordinate, the domain and range are the spread of the $x$ and $y$ values, respectively.

For instance, a complete graph of $f(x)=x^{2}-3$ is shown below.

Since the graph continues to expand in the x-direction, its domain is the set of all real numbers $\color{red}{ \lbrace x \thinspace \vert \thinspace x \text{ is in } \mathbb{R} \rbrace }$ or the interval $\color{red}{(\infty, \infty)}$ or as ausing set-builder notation . The $\color{blue}{y}$ values start at $\color{blue}{y=-3}$ and continue upward, so the range is the set $\color{blue}{\lbrace y \thinspace \vert \thinspace y \geq -3 \rbrace}$ or the interval $\color{blue}{[-3, \infty)}$.

Let's look at another example. Suppose the following is a complete graph of a function.

You might recall that solid dots are always considered part of the graph. Hollow dots, on the other hand, are used to indicate points that are not part of the graph. Here the domain is the interval $\color{red}{(-2, 2 ]}$ and the range is $\color{blue}{(-2, 4 ]}$. In set notation the domain is $\color{red}{ \lbrace x \thinspace \vert \thinspace -2 < x \leq 2 \rbrace }$ and the range is $\color{blue}{\lbrace y \thinspace \vert \thinspace -2 < y \leq 4 \rbrace}$

If the graph has asymptotes then the domain and range might require the union of several intervals. In this case, the vertical asymptotes indicate that the values $x=-1$ and $x=1$ are not in the domain.

The domain is split into three intervals that we union together: $\color{red}{(-\infty, -1) \bigcup (-1, 1) \bigcup (1, \infty)}$. The horizontal asymptote doesn't impact the range, since the center section rises above the asymptote, so our range is $\color{blue}{(-\infty, 4]}$.

Graphs of Basic Functions

Below you will be introduced to six basic functions that you should learn by heart. Use the provided graphs to look for important behaviors. Click the link below each graph to check your answers or refer to this downloadable summary.

The Identity Function

The identity function $f(x)=x$ is special because the output is always the same as the input, for example $f(5)=5$ or $f(87)=87$. Its graph is the diagonal line through the origin with a slope of $1$.

Identity Function Properties

• Domain and Range: Both the domain and range include all real numbers.
• Intercepts: Since this function crosses at the origin, both the y-intercept and the x-intercept are at the point $(0, 0)$.
• Increasing, Decreasing, Constant: The identity function is increasing over its entire domain. It never decreases or becomes constant.
• Maximums and Minimums: There are no maximums or minimums.
• Asymptotes: There are no vertical or horizontal asymptotes.

The Square Function

The square function $f(x)=x^2$ returns the square of every input, for example $f(3)=3^2=9$. Its graph is a parabola touching the origin.

With negative inputs it is best to include parenthesis like this $f(-2)=(-2)^2=4$ to get the correct values.

Square Function Properties

The graph of the square function is a *parabola*.

• Domain and Range: The domain is all real numbers. The range is the interval $[0, \infty)$.
• Intercepts: They y-intercept is $y=0$. The x-intercept is $x=0$. Both occur at the same point, $(0, 0)$.
• Increasing, Decreasing, Constant: The function is decreasing on $(-\infty, 0)$ and increasing over the interval $(0, \infty)$. It is not constant anywhere.
• Maximums and Minimums: There are no maximums, but there is a global minimum at $(0, 0)$.
• Asymptotes: There are no vertical asymptotes and no horizontal asymptotes.

The Square Root Function

The principal square root function $f(x)=\sqrt{x}$ evaluates the square root of non-negative inputs only, it is not defined for negative values.

Most calculators have a dedicated button for computing square roots. Its graph starts at the origin.

Square Root Function Properties

• Domain and Range: The domain is ${x : x \geq 0 }$. The range is ${y : y \geq 0 }$.
• Intercepts: Since this function starts at the origin, the y-intercept and the x-intercept are at the same point, $(0, 0)$.
• Increasing, Decreasing, Constant: The function is increasing over its entire domain. It is never decreasing or constant.
• Maximums and Minimums: There are no maximums, but $(0, 0)$ is a global minimum.
• Asymptotes: There are no vertical or horizontal asymptotes.

The Cube Function

The cube function $f(x)=x^3$ produces the cube of any input, which is simply that number multiplied by itself three times. For instance, $f(2)=2^3=8$ since $2 \times 2 \times 2=8$.

Most calculator do not have a dedicated button for the cube function, instead you must enter something like 2^3 to cube $2$.

Cube Function Properties

• Domain and Range: Both the domain and range include all real numbers.
• Intercepts: This function crosses at the origin, so both the y-intercept and the x-intercept are at the point $(0, 0)$.
• Increasing, Decreasing, Constant: The function is increasing over its entire domain. It is never decreasing or constant.
• Maximums and Minimums: There are no maximums or minimums.
• Asymptotes: There are no vertical or horizontal asymptotes.

The Cube Root Function

Many calculators have a command somewhere that will evaluate the cube root function $f(x)=\sqrt[3]{x}$ but it might be hard to find.

It is often easier to use the rule of exponents $\sqrt[3]{x}=x^{1/3}$ to evaluate cube roots. For example 125^(1/3) would give the cube root of $125$.

Cube Root Function Properties

• Domain and Range: Both the domain and range include all real numbers.
• Intercepts: Since this function crosses at the origin, the y-intercept and the x-intercept are both $0$.
• Increasing, Decreasing, Constant: The function is increasing over its entire domain. It is never decreasing or constant.
• Maximums and Minimums: There are no maximums or minimums.
• Asymptotes: There are no vertical or horizontal asymptotes.

The Reciprocal Function

The reciprocal function $f(x)=\frac{1}{x}$ takes any number (except $0$) as an input and returns the reciprocal of that number. The easiest way to remember what a reciprocal is, is to see a few examples.

• The reciprocal of $\frac{3}{2}$ is $\frac{2}{3}$.
• The reciprocal of $5$ is $\frac{1}{5}$.
• The reciprocal of $\frac{-6}{7}$ is $\frac{-7}{6}$.

You might recall that $\frac{1}{x}=x^{-1}$ is a rule of exponents. Because of that, many calculators have a button labeled $x^{-1}$ which will compute the reciprocal of a number.

Reciprocal Function Properties

• Domain and Range: The domain is all real numbers except $0$, since division by $0$ is undefined. The range is also all real numbers except for $0$.
• Intercepts: This function does not have any intercepts.
• Increasing, Decreasing, Constant: The function is increasing on the interval $(-\infty, 0)$ and decreasing on $(0, \infty)$. It is not constant anywhere.
• Maximums and Minimums: There are no maximums or minimums.
• Asymptotes: The y-axis is a vertical asymptote. The x-axis is a horizontal asymptote.

Looking Ahead

These six basic functions form a library that we will use to explore several different topics. Since they are building blocks for more complicated ideas, you will be expected to memorize their basic properties. This downloadable summary will help your study.