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Introduction to Modulus Functions and Graphs

Neil Trivedi

Teacher

Neil Trivedi

Contents

Modulus Function

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Modulus Graphs

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Modulus Graphs in a Higher Degree

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Practice Questions

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Modulus Function

The modulus of a value just means “its size” or “its distance from a certain point.” For example,  means the distance of  from the origin and means the distance of  from . The reason we use modulus is because if is smaller than  then the value is negative.

Definition:

The modulus of a number simply means taking the positive value.

For example, and . So, if , then or .

Example 1:

If , find:

a)

Single Step: Since the value inside the modulus function is already positive, the modulus can be ignored.

b) 

Single Step: Since the value inside the modulus function is negative, there is a need to change sign.

No answer provided.

Example 2:

Write the inequality  in the form .

Step 1: Start by rewriting the modulus function to a similar form to the inequality on the left.

We know that if , then or . So, we can say that if , then or . However, since is less than or equal to , will be between and , so, we will be left with.

Step 2: Compare the two inequalities we have and solve the simultaneous equation we obtain:

   

   

   


Therefore, we have .

No answer provided.

Modulus Graphs

First, we will look at when the modulus is around the whole function, here the form is .

We reflect up any section below the axis.

Here, we will use  and  as examples.

 

 

Two coordinate graphs compare the line 𝑦=𝑥+2 with the absolute value graph 𝑦=∣𝑥+2∣, showing how the part of the line below the 𝑥−axis is reflected upwards.

What if the modulus was only around the , which is in the form ?

We reflect any section to the right of the axis to the left.

Here, we will use and as examples.

 

 

Two graphs show that 𝑦=∣𝑥∣+2 can be formed by combining the lines 𝑦=𝑥+2 and 𝑦=−𝑥+2 to create a V-shape with vertex at (0,2).

Modulus Graphs in a Higher Degree

Even if the function is in a higher degree, we still do the same!

 Functions

 

A U-shaped quadratic curve crosses the 𝑥−axis at −2 and 4 and has a minimum value around 𝑦=−8.

A cubic-style graph rises and falls with a local maximum near 𝑦=2, crosses the 𝑥−axis, dips to a local minimum, then rises again for larger 𝑥.

The graph compares an upward-opening parabola 𝑦=𝑥2−2𝑥−8 and a downward-opening parabola 𝑦=−𝑥2+2𝑥+8, showing their shapes and where they meet the 𝑥−axis at 𝑥=−2 and 𝑥=4.

The image shows the graphs of the cubic functions 𝑦=𝑥3−2𝑥^2−𝑥+2 and 𝑦=−𝑥^3+2𝑥^2+𝑥−2, illustrating how one is the reflection of the other in the 𝑥−axis.

The image shows the graphs of the quadratic functions 𝑦=𝑥^2−2𝑥−8 and 𝑦=𝑥^2+2𝑥−8, illustrating how changing the sign of the linear term shifts the parabola horizontally.

The image shows the graphs of the cubic functions 𝑦=𝑥3−2𝑥^2−𝑥+2 and 𝑦=−𝑥3−2𝑥2+𝑥+2, illustrating how changing the sign of the leading term reflects the curve and alters its turning points.

Practice Questions

Further Practice Questions

Composite and Inverse Functions Solving with Modulus Functions