During Class - Day 6

Brain Gains

Consider the function $T(x) = \frac{1}{2}(3(x+2))^2 - 10$.

  1. Let $g(x) = c f(a(x+b)) + d$ with $a$, $b$, $c$, and $d$ constants. If $T(x) = g(x)$, which means $f(u) = u^2$, answer the following questions.
    • What is $a$?
    • What is $b$?
    • What is $c$?
    • What is $d$?
  2. Let $h(x) = A f(B(x-C)) + D$ with $A$, $B$, $C$, and $D$ constants. If $T(x) = h(x)$, which means $f(u) = u^2$, answer the following questions.
    • What is $A$?
    • What is $B$?
    • What is $C$?
    • What is $D$?

Answer

  • When $T(x) = g(x)$, we see $f(u) = u^2$, $a = 3$, $b = 2$, $c = \frac{1}{2}$, and $d = - 10$.
  • When $T(x) = h(x)$, we see $f(u) = u^2$, $A = \frac{1}{2}$, $B = 3$, $C = -2$, and $D = - 10$.

Group Meeting

Start by giving each person a moment to share what they chose to prepare for class. Help each other address any questions. When each person has had a chance to share, move on the other activities.

Practice with Functions

Compare your answers in the prep and discuss any questions. Then at the boards, tackle the following.

  • For $f(x) = x^3$, write out $f(x+2)$, $f(x-2)$, $f(x)+2$, and $f(x)-2$.
    • Use R to graph these five functions.
    • Compare and contrast them.
  • Discuss with your group, in general for a function $h(x)$, what is the difference between evaluating $h(2)$ and solving $h(t)=2$ for $t$?

Activity - Transformations of a function

Last time in class we worked with the transformation $T(x;A,B,C,D) = Af(B(x-C))+D$. Each parameter ( $A$, $B$, $C$, and $D$ ) controls some aspect of the graph of the function. Let $f(x) = 3\sqrt{x}$ and use this Desmos file to explore changing these parameters and write a description of what each parameter controls.

Open Project 1 Task 2. There are 6 functions we'll be exploring. Use the Desmos links provided there, and for each function spend time together as a group discussing how each parameter changes the behavior of the function. Write a description of what each parameter controls.

Activity - Defining and Plotting Functions in R

Run the following code in your Console to clear the environment. 

rm(list=ls())

Examine and run the following code. If you encounter any new commands, discuss together what they do, and try changing them to see how things change. 

f1 <- function(x){
sqrt(3-x)
}

f1(3)
f1(0)
f1(-100)
f1(10)

x <- seq(-10,3,0.1)
y <- f1(x)

par(mar=c(2.5,2.5,0.25,0.25))
plot(x,y,type='l')

Examine and run the following code. If you encounter any new commands, discuss together what they do, and try changing them to see how things change. 

f_quad <- function(x,a=1,b=0,c=0){
a*x^2 + b*x + c
}

f_quad(-2)
f_quad(-1)
f_quad(0)
f_quad(1)
f_quad(2)

x <- seq(-2,2,0.1)

par(mar=c(2.5,2.5,0.25,0.25))
plot(x,f_quad(x),type='l')
  • Above we defined a function with 3 parameters. Below, we define two similar functions, that differ in the first line of the function. After running these chunks of code, discuss what this difference does. 
f_quad1 <- function(x,a=1,b=0,c=0){
a*x^2 + b*x + c
}

f_quad2 <- function(x,a,b,c){
a*x^2 + b*x + c
}

f_quad1(1/2)
f_quad2(1/2)

f_quad1(1/2)
f_quad2(1/2,1,0,0)

f_quad1(0)
f_quad2(0,1,2,7)

f_quad1(0,1,2,7)
f_quad2(0,1,2,7)

f_quad1(-1/3)
f_quad2(-1/3,1,0,0)
  • Try plotting each of the two functions above with various values for $a$, $b$, and $c$ over the domain $-2\leq x\leq 2$.

Discussion - Functions, Transformations, Parameters

  1. How do we find the domain (implied domain) of a function?
  2. What is the difference between a power function and an exponential function? Give an example of each.
  3. Give an example of a transformed power function.
  4. Give an example of a transformed exponential function.
  5. Let's use Desmos and create some examples, exploring how parameters transform a function.