Class 6

Between Class Sessions - Prep for Day 6

Please spend around 2 hours working between class sessions, focusing on the tasks below. Use any extra time to complete KnewtonAlta assignments and/or work on Project tasks.

Pick something from your prep today that you can share with your group in class. It might be something new that you learned. It might be questions you have that are still unanswered, or a question along with what helped you eventually answer it. It might be something tricky that you solved. It might be a review topic that helped you remember something. It might be a conversation you had with AI that was helpful. You will have a chance to share this with your peers during class. Come ready to articulate your thinking and questions.

Preparation

(1) Practice with Functions

Let \(f(x) = \sqrt{x}\) and complete the exercise 1-4.

  1. Find the exact values.
  • Evaluate \(f(1)\)
  • Evaluate \(f(0)\)
  • Evaluate \(f(36)\)
  • Evaluate \(f(\frac{1}{4})\)
  • Evaluate \(f(15)\)
  • Evaluate \(f(-1)\)
  • Are there any real numbers that would cause us a problem when trying to evaluate \(f\) at that number?
  1. A different kind of question. Find the exact values. Assume that \(x\) is a real number.
  • Solve \(f(x) = 16\)
  • Solve \(f(x) = \frac{1}{25}\)
  • Solve \(f(x) = -1\)
  • Solve \(f(x) = x-2\)

  1. Evaluate \(f(x+7)\). Then identify the parameters in the transformation \(T(x;A,B,C,D) = Af(B(x-C))+D\) needed to obtain \(f(x+7)\). What is the relationship between the graphs of \(f(x)\) and \(f(x+7)\)?

  2. Evaluate \(f(x)+3\). Then identify the parameters in the transformation \(T(x;A,B,C,D) = Af(B(x-C))+D\) needed to obtain \(f(x)+3\). What is the relationship between the graphs of \(f(x)\) and \(f(x)+3\)?

  3. Given a function \(h(t)\), what is the difference between the symbols \(h(t) = 5\) and \(h(5)\)?

(2) Function Composition Practice

Watch the following videos from mathispower4u about Function Composition

Then complete the following:

  1. For \(f(x) = x^2\) and \(g(x) = (x+3)\), compute each of \(f(g(x))\) and \(g(f(x))\).

  2. Given the function \(f(x) = \sqrt{3-2x}\), give two function \(g(x)\) and \(h(x)\) so that \(f(x) = g(h(x))\).

Regular Reminders

Skill Practice (KA Homework)

  • Continue working on 1 – Transformations of Functions assignment

Applied Practice (Project Work)

  • Begin Project 1 Task 2.

During Class

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 calculator to explore changing these parameters and write a description of what each parameter controls.

Open Project Unit 1 - Task 2: Models and Parameter Exploration. 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 this Desmos calculator and create some examples, exploring how parameters transform a function.

Source: Class.6 on byuimath.com