\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\units}[1]{\,\text{#1}} \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus Precalculus - Graphing Basic Functions

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Single Variable Calculus
Derivatives
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Precalculus
Functions

This page explains these basic functions you need to know for calculus.

Polynomial Functions

Line (Straight)

\(y = x\)

Parabola

\(y = x^2\)

Cubic

\(y = x^3\)

Root Functions

Square Root

\(y = \sqrt{x}\)

Cube Root

\(y = \sqrt[3]{x}\)

Other Functions

Absolute Value

\(y = \abs{x}\)

Rational

\(y = 1/x\)

Once you know these functions and their graphs, you can build many other graphs using combinations and transformations. If you want a complete lecture on this topic, we recommend this video.

Prof Leonard - Graphs You Must Know [19mins-10secs]

video by Prof Leonard

Additional Graphs of Functions For Calculus

There are additional functions and graphs that you need to know for calculus. We discuss these in more depth on other pages. Here is the list with links to those pages.

Exponential \(e^x\)

     

Logarithm \(\ln x\)

     

Trig Functions \(\sin x \), \(\cos x\)

Okay, let's get started with some specific basic graphs, starting with straight lines.

Straight Line

\(y=x\)

Figure 1 [built with GeoGebra]

\(y=mx+b\)

Domain

\( \{ x ~|~ x \in \mathbb{R} \} \)

Range

\( \{ y ~|~ y \in \mathbb{R} \} \)

Line (Polynomial)

Table 1

In calculus we refer to straight lines as just lines. Other graphs are also drawn with lines but, within the context of graphing for calculus, lines usually refers to straight lines. You should be able to tell from the context if this is not the case.

Graphing straight lines is very easy. You just need two points and then you draw a straight line through them.

The equation of all lines can be written in the form \(Ax + By = C\). This is called the general form. However, the slope-intercept form is usually preferred and is the one we use most on this site. It looks like \(y=mx+b\) and most lines in calculus can be represented by this equation. The only type of line that cannot be written in slope-intercept form is a vertical line. This type of line is written as \(x=a\).

You can find more details on our equations of lines page.
Polynomial: \(y=mx+b\)
Domain: all real numbers
Range: all real numbers

Parabola

\(y=x^2\)

Figure 2 [built with GeoGebra]

\(y=x^2\)

Domain

\( \{ x ~|~ x \in \mathbb{R} \} \)

Range

\( \{ y ~|~ y \in \mathbb{R} \} \)

Parabola (Polynomial)

Table 2

The general equation form of a parabola is \(y = Ax^2 + Bx + C\). This equation is a polynomial with the highest power of two. It is ALWAYS good to write a polynomial from highest to lowest power, left to right. Another form you will see in calculus is \( (x-h)^2 = 4p(y-k) \). A parabola looks like the plot in Figure 2.

If you think of yourself as walking on the curve from left to right, the point where you stop going down and turn to go back up is called the vertex. If you know the vertex and one point on the curve, you can get the equation of the parabola. In fact, that's probably how you have learned to consider parabolas in the past. That is fine but there is a lot more to a parabola than just the vertex and the equation.

In order to get the second form, we complete the square on the \(x\) terms and combine the constants into the \(4pk\) term. You will learn a lot more about parabolas in calculus including where they come from, why the second equation given above is important and how to derive it from the general form, as well as other ways to analyze a parabola. (For a sneak peak check out our page on parabolic conics.) But for now, just make sure you know what the general equation and graph look like.
Polynomial: \(y = x^2\)
Domain: all real numbers
Range: all real numbers

Cubic

\(y=x^3\)

Figure 3 [built with GeoGebra]

In Figure 3 we show the basic cubic equation, \(y=x^3\). This equation is a polynomial with highest degree of three.
Polynomial: \(y=x^3\)
Domain: all real numbers
Range: all real numbers

\(y=x^3\)

Domain

\( \{ x ~|~ x \in \mathbb{R} \} \)

Range

\( \{ y ~|~ y \in \mathbb{R} \} \)

Cubic (Polynomial)

Table 3

Square Root

\(y=\sqrt{x}\)

Figure 4 [built with GeoGebra]

\(y = \sqrt{x}\)

Domain

\( \{ x ~|~ x \in \mathbb{R}, x \geq 0 \} \)

Range

\( \{ y ~|~ y \in \mathbb{R}, y \geq 0 \} \)

Square Root

Table 4

Up until now, we have looked at polynomials. As you will learn on the domain and range page, the domain and range of polynomials are all real numbers. This is the first non-polynomial in our list. You know from basic algebra that you cannot take the square root (also, any even root) of negative numbers in order to get a real number.
Note: We are not including complex numbers in this discussion. So we want to stick with real numbers for now.
You can see this on the graph since the graph does not extend to the left of the y-axis.
Additionally, the real number that we end up with cannot be negative (but zero is possible). On the graph this can be seen since the graph stays on or above the x-axis.
This is an important graph that you will see quite often in calculus.
Root Function: \(y = \sqrt{x}\)
Domain: all non-negative real numbers
Range: all non-negative real numbers

Cube Root

\(y=\sqrt[3]{x}\)

Figure 5 [built with GeoGebra]

Compare \(\sqrt{x}\) to \(\sqrt[3]{x}\)

Figure 6 [built with GeoGebra]

The cube root of \(x\), \(y=\sqrt[3]{x}\), is shown in Figure 5. You can see from the plot that we can put any real number in for \(x\) and get a real number out. Therefore, the domain is all real numbers and the range is also all real numbers.
The cube root looks a lot like the square root when \(x \geq 0\). However, when we plot them both on the same set of axes, as shown in Figure 6, you can see that they are similar but different. Notice that both plots go through the points \((0,0)\) and \((1,1)\).

\(y=\sqrt[3]{x}\)

Domain

\( \{ x ~|~ x \in \mathbb{R} \} \)

Range

\( \{ y ~|~ y \in \mathbb{R} \} \)

Cube Root

Table 5

Absolute Value Function

\(y=\abs{x}\)

Figure 7 [built with GeoGebra]

The absolute value graph is an example of a piecewise function. Here is another way to write the function.

\( \abs{x} = \left\{ \begin{array}{rrl} x & & x \geq 0 \\ -x & & x \lt 0 \end{array} \right. \)

If you need a refresher on piecewise functions, see our extensive discussion on this page.

\(y=\abs{x}\)

Domain

\( \{ x ~|~ x \in \mathbb{R} \} \)

Range

\( \{ y ~|~ y \in \mathbb{R}, y \geq 0 \} \)

Absolute Value

Table 6

Let's watch a short video clip discussing how to graph absolute value functions and how to visualize the distance idea on the graph.

Dr Chris Tisdell - What is the absolute value function? (part 2) [about 9min]

video by Dr Chris Tisdell

Rational Function

\(y=1/x\)

Figure 8 [built with GeoGebra]

\(y = 1/x\)

Domain

\( \{ x ~|~ x \in \mathbb{R}, x \neq 0 \} \)

Range

\( \{ y ~|~ y \in \mathbb{R}, y \neq 0 \} \)

Rational Function

Table 7

Okay, so this is a much different graph than we have discussed up until now. Take a minute to look at Figure 8 and notice a few things.
1. The graph never touches or crosses either of the axes. The graph stays either in the first quadrant (upper right) or the third quadrant (lower left).
2. Since we can never divide by zero, \(x\) can never be zero. So, the graph never crosses the y-axis.
3. Looking at the equation \(y=1/x\), we always have one in the numerator, so \(y\) can never be zero. This means the graph never crosses the x-axis.
4. The graph has two asymptotes, \(y=0\) and \(x=0\). For more information asymptotes, see our page on asymptotes of rational functions.

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