# How do I determine if this equation is a linear function or a nonlinear function?

kborgeson | Middle School Teacher | (Level 2) Adjunct Educator

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The easiest way I have for knowing the difference between linear and nonlinear is the exponent value on the variable x

It is important to understand the root word in linear.  It is LINE.  A straight line, no curves.

For example:

y = 2x - 3  This is linear because the exponent on x is one. Thus your slope is standard rise over run, like a stair step and simply goes up or down.

Y = x^2 + x + 4 is nonlinear.  When graphed it becomes a parabola, which looks like a hill on your graph.  This is because the exponent on the variable of x is more than one.

This pattern continues on for all equations.  Hope this helps to put it into easy to understand terms.  :)

dwbettine | Middle School Teacher | (Level 2) Adjunct Educator

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Here's one suggestion/method I haven't seen mentioned yet: it's called the vertical line test. This method works well for lines and curves that are already graphed. The vertical line test states that if you draw a vertical line through the line or curve in question and the vertical line intersects the line or curve in only one place, then the line or curve is a function.

One thing I did not see mentioned in any of the above posts is this: a line that is vertical IS NOT a function. So all lines of the form x = a, where a is some number, are not functions. These lines are straight, but do not have a unique x value for each value of y, therefore, not a function.

Also, understand that a line or curve can be a function without being a linear function. For example, ax^2 +by + c = 0 is a function (quadratic function) but not a linear function.

tboneson1 | Middle School Teacher | (Level 2) Adjunct Educator

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Hello Schnoop,

Determining whether your equation is a linear or non-linear function can be achieved many ways. First off, are you looking at a graph of the function, or is it an equation you are looking at? If it is a graph, then the simplest answer is "Is the graph a straight line?" You see, linear means that all your variables are only to the first power, and when variables are only to the first power, their graphs will always make a straight line. Look at the word LINEar, and notice that the first part says "LINE". That's an easy way to remember linear=line when graphed.

In equation form, is the x, y, or whatever variables you are using to the power of 1, or first power? If you are looking at the equation and the variables are only to the first power, or in other mathematical words to the first degree, then your function is linear.  In short, find all the variables and makes sure that they don't have any exponents attached to them other than a 1.

I hope this helps you, good luck :)

rakesh05 | High School Teacher | (Level 1) Assistant Educator

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The way how you differentiate a linear and a non linear function is as under-

In a linear equation the variables appear in first degree only and terms containing product of variables are absent.

e.g. y= 2x+3,

y= -3x+4,

3 y=2x-4  etc.

But in case of non linear equations at least one variable is not of the first degree or the equation contain product of variables.

e.g. y=x^2+2,

or, y^2=2x-4,

or,   y=2x+3xy-4,

or,   xy=1  etc.

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ksssy | High School Teacher | (Level 1) Adjunct Educator

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Graphically, if the equation gives you a straight line thenit is a linear equation. Else if it gives you a circle, or parabola or any other conic for that matter it is a quadratic or nonlinear equation.

ALTERNATIVE:

If the highest power of x in the equation(in x) is 1 then it is a linear equation else if the power of x is greater than 1 then it is nonlinear.(ie IF AND ONLY IF THE COEFFICIENT OF x HAVING THE HIGHEST POWER  IS NONZERO!!!)

gilbertbi | Middle School Teacher | (Level 1) Adjunct Educator

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An equation is linear if its graph forms a straight line.  This will happen when the highest power of x is "1".

Here are a few examples of linear equations:

3x + 2y = 8

y = 2x + 3

y - 2 = 3(x - 1)

(note: all variables are raised to the first degree)

Here are some examples of non-linear equations:

y = x^2 (note: x is raised to the second power)

x^2 + y^2 = 4 (note: both x & y are raised to the second power)

An equation is linear if its graph forms a straight line.  This will happen when the highest power of x is "1".

brandimarie | Elementary School Teacher | (Level 1) Adjunct Educator

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A linear equation has the following form:

y = mx + b

where

m is the slope
b is the y-intercept.

You can also perform a vertical line test. If the line touches your graphed function in more than one spot, it is not a function.

pattystephens | High School Teacher | (Level 1) Adjunct Educator

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The variable x must be either degree zero or degree 1 AND the variable y must be 1st degree in order to be a linear function.

Examples:

y = 2x - 3 (both x and y are 1st degree)

4x + 5y = 20 (both x and y are first degree)

2x - 4y = 7 + 3x (all variables are 1st degree)

y = -1  (x is degree zero and y is 1st degree; this makes a horizontal line which is a function of x)

If variable x is 1st degree but the variable y has a degree of zero, it will be a linear relation but not a function of x.

Example:

x = 4  (the graph is a vertical line and is not a function of x)

If variable y is 1st degree but the variable x has a degree other than 0 or 1, it will be a non-linear function of x.

Examples:

y = x^2 + 25 (x is not first degree)

y = 5x + 2 - x^3 (x is 3rd degree)

y = 1/x or y = x^(-1)  (x is to the power of -1)

y = sqrt(x) or y = x^(1/2) (x is to the 1/2 power; the graph is 1/2 a sideways parabola)

y = 2^x  (x is the exponent instead of the base, so the graph is exponential and not linear)

If variable y is not 1st degree, the relation will not be a function of x.

Example:

x^2 + y^2 = 4 (neither x nor y is 1st degree; the graph is a circle with a radius of 2)

x = y^2 (y is not 1st degree; this is a sideways parabola)

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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Well, if the expression of the function is like this:

f(x)=ax+b

then it is for sure a linear function.

So, all expressions, where it could be found the unknown x, but with the condition that the exponent of the unknown x, not to be bigger than 1, are expressions of  linear functions.

mathdiva | High School Teacher | (Level 1) eNoter

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A linear function is in the form y = mx + b or f(x) = mx + b, where m is the slope or rate of change and b is the y-intercept or where the graph of the line crosses the y axis.  You will notice that this function is degree 1 meaning that the x variable has an exponent of 1.  If  a function is nonlinear, then the exponent of the x variable would have an exponent of something other than 0 or 1.  Another linear function is in the form of y = a or f(x) = a, where is any real number.  The exponent of x in this function is 0 because x^0 = 1, ie y = ax^0 or y = 1x.

khewlett | Student, Grade 11 | (Level 1) eNoter

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if you have a table of values find your first differences. if your first diffs are the same it is linear. if not it is not linear.

osama402 | Student, Undergraduate | (Level 1) eNoter

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Hi,

How do I determine if this equation is a linear function or a nonlinear function?

Points:

1) If the Power Of variable is 1, so the function is linear.

Example:

Y=mx+b and Y=ax+by+C

2)Generally In mathematics a, b,c are treated as a constant and x,y,z are treated as a variable.

4) If the Power Of variable is greater than 1, so the function is not linear.It can be Quardratic Function or other.

Thanks Alot.

From:  Osama khan

crread7 | High School Teacher | (Level 1) eNoter

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I would suggest graphing it. Then it is easy to see if it is linear. But to Say that if x is to the power of 1 it has to be linear is not correct. What about log(x)? Or (x-5)/(x-7)?

jaimelesmaths | Student, Undergraduate | (Level 1) eNoter

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if you are presented with a table of values, you can determine that the function is linear if it has a constant rate of change, to test this you would take

f(xx)-f(x)/xx-x (This is supposed to be "(f of x-two minus f of x-one), divided by (x-two minus x-one)

-if you do this with multiple ordered pairs given in the table and you get the same answer each time, then you have yourself a linear function.

for example: you are given (1, 5) (2, 10) and (3, 15)

(10-5)/(2-1)= 5     and

(15-10)/(3-2)= 5

you have a constant rate of change is 5 and therefore this formula [f(x)= 5x] is a linear function

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