Question

2. Given the function k(x) = x2, compare and contrast how the application of a constant, c, affects the graph. The application of the constant must be discussed in the following manners:
• k(x + c)
• k(x) + c
• k(cx)
• c • k(x)

Answer 1

I will post a couple of examples.

Answer 2

okay :)

Answer 3

Rules of \(\large\color{black}{ \rm shifts }\) from \(\large\color{black}{ \rm f(x) }\) to \(\large\color{black}{ \rm g(x) }\).


\(\large\color{black}{ \rm f(x)=a\left| x \right| ~~~~~~~~~\rm{\Longrightarrow}~~~~~~~~\rm g(x)=a\left| x \color{blue}{ -~\rm{c} }\right| }\)
\(\large\color{blue}{ ~\rm {c} }\) units to the \(\normalsize\color{blue}{ \rm right }\).


\(\large\color{black}{ \rm f(x)=a\left| x \right| ~~~~~~~~~\rm{\Longrightarrow}~~~~~~~~\rm g(x)=a\left| x \color{blue}{ +~\rm{c} }\right| }\)
\(\large\color{blue}{ ~\rm {c} }\) units to the \(\normalsize\color{blue}{ \rm left }\).


\(\large\color{black}{ \rm f(x)=a\left| x \right| ~~~~~~~~~\rm{\Longrightarrow}~~~~~~~~\rm g(x)=a\left| x \right| \color{blue}{ +~\rm{c} }}\)
\(\large\color{blue}{ ~\rm {c} }\) units \(\normalsize\color{blue}{ \rm up }\).


\(\large\color{black}{ \rm f(x)=a\left| x \right| ~~~~~~~~~\rm{\Longrightarrow}~~~~~~~~\rm g(x)=a\left| x \right| \color{blue}{ -~\rm{c} }}\)
\(\large\color{blue}{ ~\rm{c} }\) units \(\normalsize\color{blue}{ \rm down }\).


\(\scriptsize\color{ slate }{\scriptsize{\bbox[5pt, royalblue ,border:2px solid royalblue ]{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ }}}\)

Example of a shift number 2.
\(\large\color{ blue }{\large {\bbox[5pt, lightyellow ,border:2px solid white ]{ \large\text{ }\\
\begin{array}{|c|c|c|c|}
\hline \texttt{Shifts} ~~~\tt from~~~ {f(x)~~~\tt to~~~g(x)}&~\tt{c~~~units~~~~} \\
\hline
\\f(x)= \sqrt[4]{x} ~~~~\rm{\Rightarrow}~~~~ g(x)= \sqrt[4]{x \normalsize\color{red }{ -~\rm{c}} } &~\rm{to~~the~~right~} \\
\text{ } \\
f(x)= \sqrt[4]{x} ~~~~\rm{\Rightarrow}~~~~ g(x)= \sqrt[4]{x \normalsize\color{red}{ +~\rm{c}} } &~\rm{to~~the~~left ~} \\
\text{ } \\
f(x)= \sqrt[4]{x} ~~~~\rm{\Rightarrow}~~~~ g(x)= \sqrt[4]{x} \normalsize\color{red}{ +~\rm{c} } &~\rm{up~} \\
\text{ } \\
f(x)= \sqrt[4]{x} ~~~~\rm{\Rightarrow}~~~~ g(x)= \sqrt[4]{x} \normalsize\color{red}{ -~\rm{c} } &~\rm{down~} \\ \\
\hline
\end{array} }}}\)

\(\scriptsize\color{ slate }{\scriptsize{\bbox[5pt, royalblue ,border:2px solid royalblue ]{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ }}}\)


Example of a shift number 3.
\(\large\color{ teal }{\large {\bbox[5pt, lightcyan ,border:2px solid white ]{ \large\text{ }\\
\begin{array}{|c|c|c|c|}
\hline \texttt{Shifts} ~~~\tt from~~~ {f(x)~~~\tt to~~~g(x)}&~\tt{c~~~units~~~~} \\
\hline
\\f(x)= x^2 ~~~~~\rm{\Rightarrow}~~~~ g(x)= (x \normalsize\color{red}{ -~\rm{c} })^2 &~\rm{to~~the~~right~} \\
\text{ } \\
f(x)= x^2 ~~~~~\rm{\Rightarrow}~~~~ g(x)= (x \normalsize\color{red}{ +~\rm{c} })^2&~\rm{to~~the~~left ~} \\
\text{ } \\
f(x)= x^2 ~~~~~\rm{\Rightarrow}~~~~ g(x)= x^2 \normalsize\color{red}{ +~\rm{c} } &~\rm{up~} \\
\text{ } \\
f(x)= x^2 ~~~~~\rm{\Rightarrow}~~~~ g(x)= x^2 \normalsize\color{red}{ -~\rm{c} } &~\rm{down~} \\ \\
\hline
\end{array} }}}\)

Answer 4

these are the shifts. Now I will post some stretches

Answer 5

omg i love you thank you so much for this!!

Answer 6

My latex disconnected me.

Answer 7

Sorry

Answer 8

Anyways, I will post the stretches now

Answer 9

New \(\large\color{brown}{g(x) }\) comparing the initial function \(\large\color{brown}{f(x) }\) .

Answer 10

waittt, I don't know where you're doing with those sorrryy

Answer 11

I only needed the third chart

Answer 12

But referring to the third chart, it doesn't answer the last 2 questions I was looking for :(

Answer 13

HORIZONTAL.

example 1:
\(\large\color{brown}{f(x)=x^3 }\) AND \(\large\color{brown}{f(x)=(cx)^3 }\)

horizontal stretch when \(\large\color{brown}{c>1 }\)
horizontal shrunk when \(\large\color{brown}{0<c<1 }\)

\(\scriptsize\color{ slate }{\scriptsize{\bbox[5pt, blue ,border:2px solid blue ]{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ }}}\)

example 2:
\(\large\color{brown}{f(x)=x^{3/5} }\) AND \(\large\color{brown}{f(x)=(cx)^{3/5} }\)

horizontal stretch when \(\large\color{brown}{c>1 }\)
horizontal shrunk when \(\large\color{brown}{0<c<1 }\)


\(\scriptsize\color{ slate }{\scriptsize{\bbox[5pt, red ,border:2px solid red ]{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ }}}\)
\(\scriptsize\color{ slate }{\scriptsize{\bbox[5pt, red ,border:2px solid red ]{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ }}}\)


VERTICAL.

example 1:
\(\large\color{brown}{f(x)=x^3 }\) AND \(\large\color{brown}{f(x)=c\cdot x^3 }\)

horizontal stretch when \(\large\color{brown}{c>1 }\)
horizontal shrunk when \(\large\color{brown}{0<c<1 }\)

\(\scriptsize\color{ slate }{\scriptsize{\bbox[5pt, blue ,border:2px solid blue ]{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ }}}\)

example 2:
\(\large\color{brown}{f(x)=x^{3/5} }\) AND \(\large\color{brown}{f(x)=(cx)^{3/5} }\)

vertical stretch when \(\large\color{brown}{c>1 }\)
vertical shrunk when \(\large\color{brown}{0<c<1 }\)

Answer 14

this last reply is for the stretches & shrinks, horizontally & vertically.

the charts are just for the shifts.

Answer 15

See also, for more information: http://www.math.utah.edu/~giessing/notes/ch3.6b.pdf

Answer 16

Oh ok I guess I'm not understanding this problem correctly but thanks for all the help

Answer 17

if you still need help, let me know.