During Class
Brain Gains
Many functions we use in the sciences are transformations of a basic function. For example, we can transform the power function $f(x) = x^2$ (often called a parent function) into the more complex function $g(x) = 3(x-4)^2+5$. A general form for such a transformation is $$T(x;a,b,c,d) = af(b(x+c))+d,$$ using the parent function $f$ with parameters $a$, $b$, $c$, and $d$. One of our goals this semester is to identify a parent function with appropriate parameters so that we can use the transformed function as a model for data.
Using the general transformation $T(x) = af(b(x+c))+d$, identify the parameters $a$, $b$, $c$, and $d$ needed to transform $f(u)$ into the provided function.
- $g(x) = 8(x+5)^3 - 2$ where $f(u) = u^3$
- $h(x) = 4\sqrt{2x-6} +7$ where $f(u) = \sqrt{u}$
Possible Solutions
For $g(x) = 8(x+5)^3 - 2$ we can chose $8=2, b=1, c=5, d=-2$. The function $T(x)$ using $f(u)=u^3$ is $$ \begin{align*} T(x;a,b,c,d) &= af(b(x+c))+d \\ &= a(b(x+c))^3+d \\ &= 8(1(x+5))^3+(-2) &&\text{using }a=8, b=1, c=5, d=-2 \\ &= 8(x+5)^3 - 2\\ &= h(x). \end{align*}$$ There is more than one correct answer. Another completely valid solution is $a=1, b=2, c=5, d=-2$, as $$T(x;1,2,5,-2) = 1(2(x+5))^3+(-2) = 8(x+5)^3 - 2.$$
For $h(x) = 4\sqrt{2x-6} +7$ we can chose $a=4, b=2, c=-3, d=7$. The function $T(x)$ using $f(u)=\sqrt{u}$ is $$ \begin{align*} T(x;a,b,c,d) &= af(b(x+c))+d \\ &= a\sqrt{b(x+c)}+d \\ &= 4\sqrt{2(x-3)}+7 &&\text{using }a=2, b=1, c=5, d=-2 \\ &= 4\sqrt{2x-6}+7\\ &= h(x). \end{align*}$$ It's common to guess $c=-6$ at first, but then we compute $T(x;4,2,-6,7) = 4\sqrt{2(x-6)}+7 = 4\sqrt{2x-12}+7,$$ which is incorrect. A guess, followed by an informed update, is a completely valid way to solve this problem.
Group Meeting
Project Work (Cohesive Analysis)
Compare and contrast your Project 1 Task 1 cohesive analyses. Are all plots introduced with words. Do you have sentences, outside of the code chunks, describing what is happening? If not, help each other add appropriate sentences.
Transformations of a function
- Given a general form for a transformation of a function $T(x) = af(b(x+c))+d$ and the parent function $f$, identify $a$, $b$, $c$, and $d$ for each of the transformations below.
- $r(x) = \frac{1}{4x} - 5$ where $f(u) = \frac{1}{u}$
- $t(x) = \frac{1}{2}|3-x|+7$ where $f(u) = |u|$
- $p(x) = -3x^5 -1$ where $f(u) = u^5$
- $q(x) = x^2 - 4x + 9$ where $f(u) = u^2$
- Let $f(x;a,b) = ax^b$, a generic power function. Consider the transformation $T(x;a,b,A,B,C,D) = Af(B(x-C);a,b)+D$. For each set of values for $a,b,A,B,C,D$ below, do the following.
- Identify the parent function, $ f(x; a,b) $. For example, if $a=2$ and $b=5$, then the parent function is $f(x;2,5) = f(x) = 2x^5$.
- Write down the specific function $T(x;A,B,C,D)$ produced from the given values for the parameters $A,B,C,D$.
- Adapt this Desmos file to plot and compare the two functions.
- $a=1,b=2,A=1,B=1,C=0,D=0$
Check
$ a = 1 $
$ b = 2 $
$\begin{aligned}
f(x) &= 1x^2 \\
&= x^2
\end{aligned}$
$ A = 1 $
$ B = 1 $
$ C = 0 $
$ D = 0 $
$\begin{aligned}
T(x) &= 1f(1(x-0)) + 0 \\
&= 1[1(1(x-0))^2] + 0 \\
&= 1[1(1(x))^2] + 0 \\
&= 1[1(x)^2] + 0 \\
&= x^2
\end{aligned}$
Notice $T(x)$, the "transformed function", is the same as $f(x)$, the "parent function", there is "no" transformation (or only a trivial transformation) in this case.
- $a=1,b=2,A=1,B=1,C=3,D=4$
Check
$ a = 1 $
$ b = 2 $
$\begin{aligned}
f(x) &= 1x^2 \\
&= x^2
\end{aligned}$
$ A = 1 $
$ B = 1 $
$ C = 3 $
$ D = 4 $
$\begin{aligned}
T(x) &= 1f(1(x-3)) + 4 \\
&= 1(1(x-3))^2 + 4 \\
&= (x-3)^2 + 4
\end{aligned}$
- $a=1,b=2,A=3,B=4,C=0,D=0$
Check
$ a = 1 $
$ b = 2 $
$\begin{aligned}
f(x) &= 1x^2 \\
&= x^2
\end{aligned}$
$ A = 3 $
$ B = 4 $
$ C = 0 $
$ D = 0 $
$\begin{aligned}
T(x) &= 3f(4(x-0)) + 0 \\
&= 3f(4x) \\
&= 3(4x)^2 \\
&= 3(16x^2) \\
&= 48x^2
\end{aligned}$
- $a=3,b=4,A=1,B=1,C=0,D=0$
Check
$ a = 3 $
$ b = 4 $
$ f(x) = 3x^4 $
$ A = 1 $
$ B = 1 $
$ C = 0 $
$ D = 0 $
$\begin{aligned}
T(x) &= 1f(1(x-0)) + 0 \\
&= 1f(x) \\
&= 3x^4
\end{aligned}$
Notice $T(x)$, the "transformed function", is the same as $f(x)$, the "parent function", there is "no" transformation (or only a trivial transformation) in this case.
- $a=-1,b=3,A=2,B=-4,C=5,D=-6$
Check
$ a = -1 $
$ b = 3 $
$ \begin{aligned}
f(x) &= -1x^3 \\
&= -x^3
\end{aligned}$
$ A = 2 $
$ B = -4 $
$ C = 5 $
$ D = -6 $
$\begin{aligned}
T(x) &= 2f(-4(x-5)) + (-6) \\
&= 2[-1(-4(x-5))^3] - 6 \\
&= -2((-4)^3(x-5)^3) - 6 \\
&= -2(-64(x-5)^3) - 6 \\
&= 128(x-5)^3 - 6
\end{aligned}$
- Each parameter ( $A$, $B$, $C$, and $D$ ) above controls some aspect of the graph of the function. Explore changing these parameters and write a description of what each parameter controls.
- Replace $f(x;a,b) = ax^b$ above with $f(x;a,b) = a(b^x)$ and then repeat the second exercise. Do your conclusions from part 3 still hold?
Discussion
Project 1 Task 1 discussion
- What questions do you have about the assignment?
- Are there any things you realized you needed to add after you shared your work with your peers?
- How much can we use AI?
Example Project (Cohesive Analysis)
We will complete Example Project Task 1 together and then put together a cohesive analysis to complete task 1 of the example project. We'll be building a histogram as we complete the example project. You built a scatter plot when you complete Project 1 Task 1.
- We'll open the Example Project Instructions
- Use our work on this Example Project and the feedback you receive from your peer feedback to guide in your revisions on Project 1 Task 1. Remember, you can revise and resubmit your Project 1 Task 1 today if you have found something you need to change in order to meet the specs (or if there is something you would like to update).
- Are there any questions about the expectations for the project tasks?
Functions, Transformations, Parameters
- How do we find the domain (implied domain) of a function?
- What is the difference between a power function and an exponential function? Give an example of each.
- Give an example of a transformed power function.
- Give an example of a transformed exponential function.
- Let's use Desmos and create some examples, exploring how parameters transform a function.