MAC 1105

Consider the library of functions:

𝑓1(𝑥) = 𝑥

𝑓2(𝑥) = 𝑥 2

𝑓3(𝑥) = 𝑥 3

𝑓4(𝑥) = √𝑥

𝑓5(𝑥) = √𝑥 3

𝑓6(𝑥) = 1

𝑥

𝑓7(𝑥) = |𝑥|

𝑓8(𝑥) = ⟦𝑥⟧ (greatest integer less than x) [ floor(x)].

1. Draw using DESMOS the functions 𝑔𝑖 (𝑥) = 𝑓𝑖(𝑥) + 𝑘, 𝑖 = 1, … ,9. Then,

use the slider created by DEMOS to assign different values to k. What do

you observe when the value of k changes? What is the relation between

the graph of 𝑔𝑖 (𝑥) and the graph of 𝑓𝑖 (𝑥)? Explain your conclusion with

help of a graph. (25 points)

How to use the notation: We identify each function of the library with a

subscript between 1 and 9. So, 𝑔3(𝑥) = 𝑓3(𝑥) + 𝑘 = 𝑥 3 + 𝑘 .

𝑔6(𝑥) = 𝑓6(𝑥) + 𝑘 = 1

𝑥 + 𝑘

K is a number. You can use first several positive numbers. After that a

several negative number. Observe and conclude what is the effect of

adding a constant to a function. You can simply write on DESMOS

𝑥3 + 𝑘. DESMOS will offer you to create a slider that will allow you to

consider any value for k moving the slider

Click on k

2. Draw using DESMOS the functions ℎ𝑖 (𝑥) = 𝑓𝑖(𝑥 + 𝑘), 𝑖 = 1, … ,9. Then,

use the slider created by DEMOS to assign different values to k. What do

you observe when the value of k changes? What is the relation between

the graph of ℎ𝑖 (𝑥) and the graph of 𝑓𝑖(𝑥)? Explain your conclusion with

help of a graph. (25 points)

3. Draw using DESMOS the functions 𝑞𝑖(𝑥) = 𝑘[𝑓𝑖(𝑥)], 𝑖 = 1, … ,9. Then,

use the slider created by DEMOS to assign different values to k. What do

you observe when the value of k changes? What is the relation between

the graph of 𝑔𝑖 (𝑥) and the graph of 𝑓𝑖 (𝑥)? Explain your conclusion with

help of a graph. (25 points)

4. Describe the figure of the cover page using inequalities involving the

absolute value function. (15 points)

5. Unfold your imagination and surprise with an interesting drawing using

DESMOS. (10 points)

You will need to become familiar with DESMOS. If you need any help do not

hesitate to ask me.

Now, enjoy your work!

Use this button to change the values of k