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An in-depth exploration of modulus functions, including their definition, properties, and methods for sketching their graphs. It covers various examples and uses both the definition method and graphic calculators to illustrate the concepts.
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Modulus functions and their graphs. luxvis.com 11/12/
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Modulus Functions
The modulus function or otherwise known as the absolute value of a real number x is defined by the following
if 0 if 0
x x x x x
It may also be defined as x x^2
Properties of the Modulus Function
Property Example The absolute value of x is written as |x|. It is defined by the following:
( ) a 0 | | ( ) a <
a if a a if
x can be thought of as the distance that x is from zero. For example, the distance that 5 is from zero is 5, whereas the distance that - 3 is from zero is 3. So we can then say 5 ^5 whereas^3 ^3
If a and b are both non negative or both non positive
Then we get the following expression a x a
k a x k a
Then we get the following expression 2 a x 2 a
a a b b
Sometimes x is referred to as magnitude of x , or the modulus of x , which can be thought to roughly mean the size of x
Now we sketch the graph Be careful when sketching the graphs
Notice how the graph is positive for all values of x
Now we could have just used our graphics calculators and we would had obtained the above graph quickly, however it is important to be able to do the maths.
Let us look at a few more examples on using modulus functions. Example 2: y x 2
Start with the definition always We use the definition of
( ) a 0 | | ( ) a <
a if a a if
to see how to sketch this modulus function
Now replace the numbers with what we actually have
Here in the place of a x 2
So we have the following
x if x x x if x
Now separating the two expressions into two y x 2 if (^) x (^2) 0 which basically means x 2
The other expression becomes (^) y (^) x (^2) x (^2) , which applies for the following x (^2) 0 x 2
Now we sketch the graph once again using both expressions
Notice how the next graph looks a little different; in the sense the modulus signs only cover the x values
Example 3: Sketch y x 4
Definition as always Remember the definition a 0 | | a <
a if a a if
Now use the expressions for our question 4 x 0 | | 4 -x <
x if x x if
Now do the first expression (^) Now the graph of y x 4 is for x 0
Now the other expression (^) While the graph of y x 4 if for x 0
Step- 4 :Press the absolute button and input the equation directly into the calculator
Step- 5 :Press Enter the button and you will get the following. And drag the equation to the next line
Step- 6 :Now press the graph button on the top of the screen and you will get the following
Step- 7 :drag the equation to the screen on the bottom
Step- 8 :Now click on the screen below and resize and you get
Step- 9 :That is your graph You could had also done it by directly input the function from the screen using the short word abs( x)- 4 and moving to the graph quickly
Skill Builder Try the following questions to hone your skills Sketch the graph of each of the following modulus function. Make sure you include the domain and range of the function, a) Sketch the graph of y^ ^ x ^5 b) Sketch the graph of y^ ^ x ^2 c) Sketch the graph of y x 1 d) Sketch the graph of y x 1 5 e) Sketch the graph of y x 3 5 f) Sketch the graph of y 2 x 1 5 g) Sketch the graph of y^ ^4 x h) Sketch the graph of y^ ^ x ^2 ^1
More difficult questions regarding modulus functions
How do we sketch the following graph? y x^2 x 2 Let us use the definition of modulus
a 0 | | -a <
a if a a if
2 2 2 2 2
x x if x x x x x x if x x
Now this is where it gets difficult x^2^ x 2 0
We can factorise the above quadratic equation, x^2^ x 2 x 2 x 1
Now lets us sketch the normal graph of y x^2^ x 2
The normal graph Notice the how the bottom is reflected in the x-axis
Method 2: Simply sketch the graph. y x^2^ x 2 The absolute value of a positive number is equal to that number, and the absolute value of a negative number is equal to the negative number and is therefore positive. So we simply sketch the normal graph and reflect in the x-axis the part of the graph that has a negative y value.
Example: Sketch the graph of y (^) 2 x (^) x (^1) x (^3) Let’s have done it the fast way
And if you reflect the
Skill builder
a) Sketch the graph of y x^2^ 3 x 10 b) Sketch the graph of y x^2 2 x 8 c) Sketch the graph of y (^) x (^1) x (^1) x (^3) d) Sketch the graph of y (^) x (^1) x (^3) x (^4) e) Sketch the graph of y (^) x (^1) 2 x (^3)
f) Sketch the graph of y (^) x (^2) ^2 x (^2)
SOLVING MODULUS EQUATIONS
We normally use the absolute value in Physics when we are looking at the magnitude of something which basically means the value without worrying about it direction like in the case of velocity. Simply put the absolute value means how far the number is from the zero on a number line.
6 its absolute value is 6 6 Its absolute value is 6 also , as it is 6 units away from zero on the number line.
So in practice the 'absolute value' means to remove any negative sign and give its positive value.
Example
Problems Answer | 4| 4 |-12| 12
Absolute values means How far a number is from zero In Physics we tend to use the expression the magnitude of a number , it size So if we are asked to find the absolute value of an number we just give a positive value
For example the absolute value of - 30 is 30 Absolute value of - 5 is 5 We tend to ignore the negative and just state the number To show the absolute value To show we are talking about the absolute value of a function we use the following |x|, we call them bars
Sometimes calculators we show the absolute values be using the expression abs(-1), which means find the absolute value of - 1
Sometimes absolute value is also written as "abs()", so abs(-1) = 1 is the same as |-1| = 1
Case 1- Questions involving < inequality or the inequality
The solution is always an interval and the pattern holds true.
The inequality to solve What it means x a a x a x a a x a
Examples involving case 1 type of problems
Solve the following inequality x 5
This means that the solution is giving by the above definition 5 x 5 and that is your answer However if you wanted the complete working out you will need to do the following: We split it to put parts from the definition of the modulus which is namely the following ( ) if 0 ( ) if 0
x x^ x x x
x 5, provided x 0 And the next part of the equation becomes 5 5 provided 0
x x x
Now we have the two parts and we get the following inequality that provides us with the solution that is: 5 x 5 Now we could write this as an interval as 5,5
Another example to illustrate the basic idea is solve for x the following
Let us use the quick way of solving this inequality It is of the form of case 1, so we get the following expression
2 x 3 8 8 2 3 8 8 3 2 3 3 8 3 5 2 11 5 11 2 2
x x x x
I could had separated the expressions and proceed with two equations but I decided in using the fast method to obtain the answer
Case 2- the inequality is > or
x a
x a or x a
Remember that
( ) if 0 ( ) if 0
x x x x x
So that means x a x a
x a x a or x a
So the solution is two inequalities not one. Do not try to combine them into one inequality as this would be a mistake.
Example
Solve x 2
Solution x 2 and x 2
Now we can write this as , (^2) (^) 2, It would be a mistake to combine this and write is as 2 x 2? Why? You cannot have x 2 and x 2 Hold true at the same time.
Solve 2 x 3 5
First step solve this equation 2 3 5 2 2 1
x x x
Then next step solve this equation 2 3 5 2 8 4
x x x
And the answer is x 1 x 4
MODULUS FUNCTIONS
1 What is modulus?
It can be thought of as distance from zero without worrying about positive or negative
( ) a 0 | | ( ) a <
a if a a if
2 What is the meaning of |3|?
It is called the modulus of 3 or the magnitude of 3 or the size of three which is 3 of course
y | x 1|^ use the definition and then rewrite it as ( 1) 1 0 | 1| ( 1) 1 0
x x x x x
Now split the problem into two , first do the top part y ( x 1) x 1 0 and then do the second bottom part y ( x 1) x 1 0
3 How do you sketch graphs of modulus
Essentially you will need to split the graph up using the definition and then sketch each separately
4
What do you need to be careful when working with modulus?
Watch out for the inequality signs carefully. If you multiply or divide by a negative number Also include the brackets!
Be careful to use a number line to help you with the domains and ranges also!
