普通的非线性函数

Nonlinear Functions

Before we get into some common nonlinear functions, make sure you understand what nonlinear functions are and what makes them different from linear functions.

When graphed, nonlinear functions are not straight lines (i.e., they are non linear).

Nonlinear functions are those that DO NOT follow the form:

$$y=mx+b$$

So, functions that have a term where $x$ is part of a complicated operation (like $x^2$, $x^3$, or $\frac{1}{x}$) are examples of nonlinear functions.

On this page, we'll look at three of the most common nonlinear functions: $x^2$, $x^3$, and $\frac{1}{x}$.

Functions with $x^2$

Functions with the term $x^2$ have a distinct U-shape when they are graphed. This shape is called a parabola. Think of it like a smile or a rainbow!

Notice that when the function includes the term $x^2$, the parabola faces up. When the function includes the term $-x^2$, the parabola faces downward.

The term in the function that doesn't include $x$ (i.e., the naked/plain number) tells us where the parabola crosses the y-axis.

The function graphed above is $y=x^2+3$. Because the naked/plain number in the function is $3$, we know that the parabola crosses the y-axis at $y=3$.

The coefficient of the $x^2$ term in the function determines how narrow or wide the parabola is. The larger the coefficient is, the narrower the parabola will be.

Functions with $x^3$

Functions that include the term $x^3$ have a double J-shape when they are graphed. Imagine drawing a squiggly line in the middle of the graph.

As with the $x^2$ functions, the sign in front of the $x^3$ determines which way the shape of the graph goes. Here is a graph of $y=-x^3$.

It grows faster than a straight line (linear) or a U-shape (quadratic).

Functions with $\frac{1}{x}$

Functions that include $\frac{1}{x}$ (or any form of that fraction where $x$ is in the denominator), have an L shape, like a swoop.

There are two symmetrical swoops per graph, and two asymptotes. An asymptote is a place on the graph where the line absolutely cannot pass.

Because $x$ is the denominator of a fraction, there is a vertical asymptote at $x=0$. That means that there's no possible value for $x=0$ (because you can't divide by $0$)!

(There's also a horizontal asymptote at $y = 0$, but you don't have to worry about that yet!)

Notice how both swoops approach, but never touch, $x=0$. This is the location of the asymptote.

As with the other nonlinear functions, when we make the leading term negative, the graph changes. It flips over diagonally, but the asymptotes remain the same.

There are other factors that can change where all of these graphs appear on the plane, but the shapes will always stay roughly the same!

To understand more complicated functions, it helps to start with parent functions. A parent function is the simplest form of a function.

Parent functions have basic shapes and behaviors. We can change these shapes by stretching, shrinking, or shifting the graph.

By learning parent functions first, it becomes easier to understand how more complex functions are built.

In this lesson, we’ll explore the most common parent functions and learn how they work as the foundation for many other types of equations. Once you understand these basic shapes, you’ll have a better grasp on how more complex functions behave!

Parent and Common Functions

We are going to look at parent functions, which are basically the main function in which the rest of the family of functions is derived from. This lesson will exclusively cover algebraic functions, but you will soon learn about:

• rational functions
• inverse functions
• exponential functions
• trigonometric functions
• logarithmic functions
• polynomial functions
• radical functions

If you are interested more broadly in polynomial transformations, you can learn those, as well. It's a similar concept, while also includes combining transformations.

Quadratic Functions

The quadratic parent function is written as $f(x) = x^2$, and its graph is a smooth, U-shaped curve called a parabola. The vertex is the point $(0,0)$, which is the origin, and it opens upward. As the value of $x$ gets larger or smaller, the value of $f(x)$ increases rapidly, showing the function's symmetric and curved nature.

Cubic Functions

The cubic parent function is written as $f(x)=x^3$, and its graph has an S-shape, with one curve rising and the other falling. The origin is the point of symmetry, and the function passes through the origin. Values become more positive or negative as $x$ moves away from zero. Unlike the quadratic function, the cubic function can have both positive and negative values for $f(x) and continues to rise or fall without bound.

Exponential Functions

The exponential parent function is written as $f(x)=b^x$, where $b$ is a positive constant greater than one. Its graph shows rapid growth or decay, depending on whether $b$ is greater than $1$ (growth) or between $0$ and $1$ (decay). The graph always passes through the point $(0,1)$. As $x$ increases, the function grows quickly. While $x$ decreases, the function approaches zero but never actually reaches it.

Absolute Value Functions

The absolute value parent function is written as $f(x)=|x|$, and its graph forms a V-shape with a sharp point at the origin $(0,0)$. The function outputs only non-negative values, meaning it reflects any negative inputs to positive values, creating two straight lines that open upward. The graph is symmetric around the $y$-axis, with both sides having the same steepness.

Trig Functions

The parent trigonometric function, like $f(x)=sin(x)$ and $f(x)=cos(x)$, have wave-like graphs that repeat in regular intervals. The sine function starts at the origin and oscillates between $-1$ and $1$, while the cosine function starts at $(0,1)$ and also oscillates between $-1$ and $1$. Both functions have a period of $2\pi$, meaning they repeat every $2\pi$ units along the $x$-axis, making them perfect for modeling cyclical behaviors like sound waves, tides, or seasonal patterns.

Looking Beyond

If you want to consider the implications beyond Algebra $2$, then you can also look into transcendental functions.

(Video) Graphs of Non-Linear Functions

by Larry Schmidt

In this video, LarrySchmidt describes four types of common nonlinear functions and how to recognize them. The functions he describes are:

Quadratic functions: $f(x) = x^2\\$ Cubic functions: $f(x) = x^3\\$ Square root functions: $f(x) = \sqrt{x}\\$ Absolute value function: $f(x) = |x|\\$

The functions he describes are in their basic forms. They match the equations in the list exactly.

When equations become more complicated, the graphs can shift on the coordinate plane. They can also be stretched or compressed up, down, left, or right.

Even when this happens, the overall shape stays the same. A quadratic function will always look like a “U” or an “∩”. An absolute value function will always have a sharp V-shape, just flipped or moved.

These patterns help you recognize functions, even when the equations look different. Watch the video, then try this next question!

This is a small snippet of the middle of the graph of a cubic function. the equation for the graph is:

$$f(x) = \frac{((x-3)^3 + x)}{20} + 4$$

As you can see, it can get messy! But the most important thing is that the highest exponent is $3$--that's what determines the basic shape of the graph.

Visualizing a few types of common functions $\\$

We've learned about the linear function. Here is a graph of one:

You can spin it, move it, squeeze and it will still be a linear function. Linear function is a type of common function.

There are a lot of different types of function and they are used to describe different relationships. Here is a chart of a few common functions, you will be seeing them more in the future. Observe how they are different.

If you are interested, the chart below also shows the graphs and expressions of these functions.

To test your understanding, take a look at the function below:

Graphs as Bulls

There are some really fun ways to remember the shapes for parent functions. This image of bulls is a great way to memorize some of the common parent functions!

Imagine these bulls the next time you're thinking about parent functions! Parent functions are like the starting point for all other functions. They're the simplest examples.

Linear functions are the easiest type of function to understand. They always make straight lines. Their equation is like a recipe: $y=mx+b$. The $m$ tells you how steep the line is, like a hill. The $b$ shows where the line starts.

A quadratic function makes a U-shaped graph called a parabola.

The cubic function has an S-shaped curve, looking like a squiggly line. It grows faster than a straight line or quadratic function.

Exponential functions grow really, really fast - faster than straight lines or U-shapes. Their graph gets really close to the $x$-axis but never actually touches it, like a plane that's about to land but never does.

The absolute value function makes a V-shape. It's like drawing two straight lines that meet at a point. It flips any negative number into a positive one. For example, $-3$ becomes $3$!