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

Expert Answers
pattystephens eNotes educator| Certified Educator

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.


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.


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.


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.


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)


tboneson1 eNotes educator| Certified Educator

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 :)

dwbettine eNotes educator| Certified Educator

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.

gilbertbi eNotes educator| Certified Educator

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".


rakesh05 eNotes educator| Certified Educator

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.


ksssy eNotes educator| Certified Educator

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.


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!!!)

brandimarie eNotes educator| Certified Educator

A linear equation has the following form: 

y = mx + b 


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. 

kborgeson eNotes educator| Certified Educator

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.  :)

giorgiana1976 | Student

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


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 | Student

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.

a123personiam | Student

The x has to have a power of 1

khewlett | Student

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


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



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


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 | Student
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

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

sandrabaldwin | Student

If an equation is a linear function then when graphed, it creates a line.   Solve the equation so that it takes the form y=mx + b.  m is going to be the slope of the equation and show how steep the line is.  b will tell you where the y intercept is. 

If the equation will not simplify into the y=mx+b form, it is not a linear equation, or you have made a mathematical error.  Linear equations will never be raised to a power ie: x2 once simplified.  Don't let an equation that IS raised to a power trick you though!  Be sure to see if any terms cancel themselves out before you judge it to be linear.

To graph an equation, make a chart with your x and y values.  Plug in values for x and then solve the equation and obtain the y value.  Finally, chart the results on a coordinate grid to show the path of the line. You only need 2 points to make a line, but a third point is advised just to make sure you've really gotten a straight line!

Happy mathing!  Good luck! 


agvickers | Student

probably the easiest way for me is to plug in the equation on a graphing calculator and see if its linear

ashifs | Student

all linear functions are in the form

ax+by = c  where a,b,c are real values and x,y are variables.


rksudhir | Student

a function like

f(x)=ax+b is a linear function

here expression 'x' have power=1

functions like


are non linear functions

rksudhir | Student

linear means single

when a function is like


then it is called a linear function

rksudhir | Student



then it is linear function becasue here expression 'x' raises to power '1' only such functions are linear functions and others where exponent 'x' have power greater than one these are non linear functions

mitodoteira | Student

Find all the variables.  If no variable has a power of >1, the function is lenear.

azzyj | Student

if it uses: y=mc+c, it is linear. A non-linear function will include powers/indices 

goldsmith077 | Student

Its a single line


kanishkporwal | Student

Any equation having both the the degrees of both the variables as one can be called as a linear equation.Otherwise its a non linear equation.

However sometimes you encounter problems having dy/dx & dx/dy instead of  x & y these equations are called as differential equations.


cheezea | Student

linear function: y=mx+c, x has power 1

nonlinear: x has a power >2, for example, (x+a)(x+b) where if you expand it it becomes x^2 + bx + ax + ab

abc10150 | Student

if the x is to the power of one, it is linear(can be written in the form y=mx+b).

unless it Is a absolute value equation (y=|x| or some variation there of)

PS the vertical line test shows when something is a function, not whether it is linear or non-linear

dareg17 | Student

A good way to tell if it is a linear function or not is to graph it.  After graphing it, use the verticle line test to decide wether not it is a function.  draw a line from top of the graph to the bottom, if the line crosses your verticle line more than once it is not a function.

another way to decide if it is or not is to make a table. if two x values are the same it is not a function.

snickel1 | Student

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smiley19 | Student

you would graph your function first.

If it is a straight line then you know its a linear function.

But if it does not make a straight line it is a nonlinear function.

The best way is to use a ruler after you graph and see if its straight.

tallknight | Student

when trying to determine if a given function is linear our not, look at the exponetial of the unknown.  If it is NOT 1 then is is NOT linear.  If it is exactly 1 then it is linear.  

E.g. f(x) = a*x+b is linear.  (no listed exponent, so we take it to be 1)

F(x) = a*x^2-b*x+c would be non-linear.

burucool123 | Student

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


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.

anon1234513 | Student

If y=x+c it is linear regardless of the gradient.

If y=x^z+c it is not a line. Hence the term non-linear. y=x^2 is a parabola. 

vaaruni | Student

There are two ways by which we can test if a given equation is a linear function or not .  

1)  By Graph :  I the graph of the equation is a straight line then the function is a linear function.

ii) By form of the equation : If the degree of the equqtion is one i.e. The power or the exponent of the variable is 1 then the equation is linear.  In this case the equation will be in the form :  y = mx + c [ where m is the slope of the line and  c is the y- intercept ] .  

rajesh66 | Student

Linear equation means equation that plots straight lines on graph.It is of the form ax+by+c=0 where a,b,c are real numbers and x and y variables.

If it can fit the form Ax+By=C it is a linear function, this means no exponents in this form.

morthanian | Student

VLT, the vertical line test. Draw a line verticly through all the points graphed. If two points are on the same line, the entire equation isn't a linear function

cheezea | Student

* more than or equal to 2 (not >2)

ericmilamattim | Student

Linear function is a 1st degree function, meaning the exponent of x is 1. Any exponent for x such as negative, positive that is greater than 1 is not linear.


y=2x + 1, since the exponent of x is 1 it is a linear function

y= 2x^3, since the exponent of x is 3 it is not linear function

y = 4, it is considered linear, since we can rewrite this as y =0x + 4

y=1/x  + 2, it is not linear since x is at the denominator, we expresed in standard form it will become y =x^-1 + 2

y= 2x^(2/3) is not a linear since it has a fractional exponent that is not equivalent to 1.

wescarlyon337 | Student

All linear functions are in the form

ax+by = c  where a,b,c are real values and x,y are variables.

Does this help?


etjoslin | Student

It's simple: The basic form of a linear function is f(x) = ax + b. 

a = the constant

x = the variable

b = the y value

If a is not raised to any power greater than one, than the function is linear. However, for example, if the equation is:

f(x) = ax^2 + b

than the equation is no longer linear, but instead quadratic. It moves up from there, but as long as there are no exponants, your function is linear!

courtypoo111 | Student

I really do not get how to explain it, so here is an example and a worksheet that you can print out and pr ctice with!!  You are welcome!!  :)

tadertot246 | Student

If the rate of change is constant, then its linear but if its not constant, then its nonlinear.

jishnudeepkar | Student

it is not necessary that it has to be eqn in one variable.

a general 2nd degree eqn like ax^2 + by^2 + 2hxy +2gx +2fy+c=0

represents any curve( line, circle, para hyper bola etc). even if

ax^2 + by^2 =0 then also the eqn is second degree as max power is 2 in 2hxy where poer of x and y is one and1+1=2.

if even 2hxy=0 u can easily see that it becaomes a st line eqn.

quantatanu | Student

if the variavle of the function has power 1 than the function is linear,


that is suppose the function is f(x) = a x^n + b x^m + ... 

you check, if at least one of the power of "x" in the whole expression if not "1" then its non-linear, but if all the powers of "x" are either "0" or "1" then its linear.



1) f(x) = a x + b = a x^1 + b x^0, power of x are 1 & 0 so linear.

2) f(x) = a x + b x^(0.5), powers of x are 1 & 0.5 so non linear.

3) f(x) = a x^100 + b, powers of at least one of the x is 100 =/= 1 so non linear.

4) f(x) = a, power of x is "0" so linear. 

tonyz98 | Student

the exponent value of x.

hngould17 | Student

A linear function will be a straight line with an x and y intercept and the  same slope through the whole line

A nonlinear function will not be a straight line and will have an indefinite  slope.

overworked | Student

Linear functions come in the form y=ax+b

They are straight lines.

Quadratic functions come in the form y=ax^2 + bx + c

They are parabolas.

kaj12345 | Student

check out if the power of x is one

shihan | Student

* more than or equal to 2 (not>2)

jess1999 | Student

There are actually multiple ways to check if an equation or graph is a linear function or not . First make sure that graph fits the equation y = mx + b . y = the point for y ; x = the point for x ; m = slope ; b = y intercept . By using this equation you'll be able to tell if it is a linear line or not . You can also look at the graph to see if it has any curves at all . A linear line must not have any curves at all in the graph . 

kevin0001 | Student

The way to tell if an equation is linear is by using y = mx + b. M and b can stand for any number as long it stays the same number for one equation. If the equation has a structure like this, then the equation is always linear. If an equation also has a structure like this : mx = y, then the equation is linear. These equations are linear because they grow at a constant rate. One way to tell if an equation is nonlinear, is if it has an exponent in it. Since exponents grow at different rates, the equation would not be linear.

nisarg | Student

if the x in the equation is not squared or multiplied by its self. so linear looks like x+any number=        where non linear is x^any number

maria-vivanco | Student

If an equation is linear when graphed it will display a line going either upward or downward. the equation itself wouldn't  have exponents and the only exponent it would have is to the power of 1. 

a nonlinear function would show a graph that isn't a line or a parabola. The equation would include exponents. 

lv1997 | Student

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

If the equation has an x value to the exponent one, it is a linear equation. For example:

y = 2x + 5

If the equation has an x value to the exponent two, it is a parabola. For example:

y = 3x^2 +5x + 3

crystaltu001 | Student

If x is by itself or equals to 1  then it is a linear function and for it to be non linear then the line on the graph is curved

parama9000 | Student

If the graph plotted isn't in  the form of a curve, then it is a linear graph, and thus a linear function.

rachellopez | Student

A linear function is just a straight line. The general formula for a linear function is represented by the equation y=mx+b. This type of equation is set in slope-intercept form, which is the most common form you will see. M represents the slope, or how steep the line is, and B represents where the line crosses to y-axis, or the y-intercept. An example of a linear equation would be y=2x+1. A nonlinear function would be anything that is not a straight line.

A function is usually linear if it follows a similar format of 

y = mx +b. what you have to notice here is the X, if the X has no exponents on it, then it is usually linear, but if the X is X squared or higher then it would not be a linear function.``

` `

rimmery | Student

What makes a function linear or non-linear is by the exponent value on the variable x. 

linear function has x with an exponent value of 1. This is a line

A non-linear function has exponent value of x > 1.

For example, consider `y=x`

x has an exponent value of 1. The graph is a straight line through the origin. This graph is linear. 

Now consider `y=x^2`

x has an exponent value of 2. The graph is a parabola and not a straight line, and is therefore non-linear.

zumba96 | Student

A linear equation has the following form: 

y = mx + c 


m is the gradient 
c is the y-intercept. 

atyourservice | Student

y=mx+b form is linear and anything other than that is nonlinear

bloopyone | Student

This only works if you have graphed the equation, then use a Pencil Test to figure out if its a function or not.


Line the pencil vertcially with the graph (parellel to Y-axis).

Run the pencil across the graph.

IF, at any point, the your graph touchs the pencil twice, then its NOT a function. If it doesnt, then it is a function.

redrose817 | Student

If the function is in the form of y=mx+b  (m(slope)& b=constant ie. 1,2,etc)  then its linear 

whereas if its in y=ax^2 + bx +c : Quadratic   

ax^3+bx^2+cx+d= Cubic 

so all functions if they are not in the form of y=mx+b or y=5x+2 then they are not linear 

for ex- y=4/x is not linear  wherea y=4x is linear for c=0

rei1213 | Student

If it is a linear equation, the highest power the you can see is 1 For example: y=2x-3, 3y=10x+8

non-linear equation can be quadratic, cubic, quartic equations

oldnick | Student

a linear equation  is an application  f : R -----> R ( R reals field)

x, y `in` R `rArr`  f(x) - f(y) = k ( x - y)  where k is a costant.





parama9000 | Student


M: the gradient

C: the y intercept 

If you see an equation in this form or are given similarly related values then it is.most probably a linear function.

If the x is by itself then it is a linear function like in the case of y=mx+b case where x has no exponents if x does have exponents besides one then it is not linear.

xdlovey619 | Student

To determine you have a linear equation first try your best to get the equation in the following form:

it is typically in the form of y=mx + b

m=slope (rise/run), needs to be constant

b=where the line intersects the y-axis

Then look at your exponent.

The exponent of the variable (x) in an equation tells you the type of equation it is. a variable to the 1st power means the equation is linear, a variable to the 2nd power is a parabola, etc.