Linear Transformations Vertical and Horizontal Stretching and Compressing Examples
Summary
TLDRThis video explains how to horizontally and vertically stretch functions. It walks through examples, first demonstrating how to horizontally stretch a function by a factor of two and discussing the effect on the slope, which decreases. The presenter also shows how to graph the transformations and update the function’s equation. Next, the video covers vertical stretching by a factor of two, explaining how to multiply the entire function by the stretch factor, and showing both the graphing and notation methods to find the new function equation.
Takeaways
- 🧠 Horizontally stretching a function by a factor of two reduces the slope.
- 🔢 Horizontal stretching affects only the X portion of a function.
- 🔀 You can either divide the X by 2 or multiply it by 1/2 for horizontal stretching.
- 📉 After horizontally stretching, the slope of a line decreases (e.g., from 3 to 3/2).
- 📝 Graphing helps visualize how stretching changes the shape of a function.
- 📈 When stretching horizontally, multiply each X-coordinate by the stretch factor.
- 📊 Vertically stretching by a factor of two increases the slope and makes the function taller.
- ⬆️ Vertically stretching changes the Y-coordinates of the function.
- ✏️ Multiplying the entire function by the stretch factor applies vertical stretching.
- 📐 Using both graphing and notation helps reinforce understanding of function transformation.
Q & A
What is the main focus of the video?
-The video focuses on stretching and compressing functions, specifically discussing how to horizontally and vertically stretch a function and how to write the new equation after these transformations.
What happens when a function is horizontally stretched?
-When a function is horizontally stretched, the slope of the function decreases, making the function appear 'flatter.' The x-values are affected, and this results in the graph stretching outward horizontally.
How do you horizontally stretch a function by a factor of 2?
-To horizontally stretch a function by a factor of 2, you need to divide the x-variable by 2. Alternatively, you can multiply the x-variable by 1/2, as both approaches yield the same effect.
What impact does horizontal stretching have on the slope of the function?
-Horizontal stretching reduces the slope of the function. For example, if the slope is initially 3, it becomes 3/2 after a horizontal stretch by a factor of 2.
What does the notation 'G(x)' represent in the video?
-In the video, 'G(x)' represents the transformed function after applying either a horizontal or vertical stretch to the original function 'f(x).'
How can you visually check if a horizontal stretch was applied correctly?
-You can visually check by graphing both the original function and the transformed function. The transformed function should look stretched horizontally, with the x-coordinates of key points multiplied by the stretching factor.
What is the key difference between horizontal and vertical stretching?
-Horizontal stretching affects the x-values of the function and makes the graph wider, while vertical stretching affects the y-values, making the graph taller or shorter depending on the factor.
How do you vertically stretch a function by a factor of 2?
-To vertically stretch a function by a factor of 2, you multiply the entire function by 2, which scales the y-values by 2 and makes the graph taller.
What happens to the slope when a function is vertically stretched?
-When a function is vertically stretched, the slope increases. For example, if the initial slope is 3, it becomes 6 after a vertical stretch by a factor of 2.
What are two methods mentioned in the video for finding the transformed function?
-The video explains two methods: using notation (mathematically altering the function) and graphing (visually transforming the function by adjusting the coordinates and then plotting the new points).
Outlines
📐 Understanding Horizontal Stretching of Functions
In this section, the speaker introduces the concept of stretching and compressing functions, with a focus on horizontally stretching a function by a factor of two. The key idea is that when horizontally stretching, the slope of the function decreases, and this transformation affects only the X-coordinate. The speaker explains how to represent this change in notation and graphically, emphasizing that multiplying the X-coordinate by 1/2 has the same effect as dividing by 2. They illustrate this process by modifying the function and graphing the changes, showing how the slope changes from 3 to 3/2.
🖊 Visualizing Horizontal Stretch through Graphing
The speaker continues the explanation by graphing the original function and its horizontally stretched version. They show how to adjust the X-coordinates by multiplying them by 2 to reflect the horizontal stretch. By comparing the original and transformed functions, they demonstrate how the graph visually represents the horizontal stretching, with key points shifting on the X-axis. The speaker notes that the Y-intercept remains the same while the slope changes, reaffirming that the slope decreases as the function stretches horizontally.
📏 Vertically Stretching Functions by a Factor of Two
In this section, the focus shifts to vertical stretching, where the speaker describes stretching a function vertically by a factor of two. They graph the original function, and then explain how vertical stretching affects the Y-coordinates. By multiplying the Y-values by 2, the graph becomes taller, with steeper slopes and more negative Y-intercepts. The speaker shows that a vertical stretch increases both the slope and Y-intercept, changing the equation from 3x - 2 to 6x - 4.
📊 Applying Vertical Stretching Using Notation
To reinforce the concept, the speaker demonstrates how to apply vertical stretching through notation. They explain that vertically stretching a function means multiplying the entire function by a constant factor, in this case, by 2. This transformation results in multiplying both the slope and the Y-intercept by 2, leading to a new equation for the transformed function. The speaker concludes by summarizing the process of vertical stretching and showing that it can be understood both graphically and through algebraic manipulation.
Mindmap
Keywords
💡Horizontally Stretch
💡Vertically Stretch
💡Slope
💡Factor of Two
💡Transformation
💡X-coordinate
💡Y-coordinate
💡Graphing
💡Notation
💡Y-intercept
Highlights
Introduction to stretching and compressing functions, focusing on both horizontal and vertical transformations.
Horizontal stretches affect only the x-coordinates of a function, causing a decrease in slope as the function appears 'stretched out'.
Key concept: Horizontally stretching a function results in the slope decreasing, as shown by graph transformations.
Demonstrating that to horizontally stretch by a factor of two, the x-portion of the function is divided by two or multiplied by 1/2.
When transforming a function horizontally, the x-coordinates of points are multiplied by the stretch factor to plot the new function.
Graphing approach: Horizontal stretching by multiplying the x-coordinates of specific points on the graph (e.g., 2 becomes 4, 1 becomes 2).
Visual demonstration of horizontal stretch: Observing how the function's slope decreases when plotted after the transformation.
Comparison of initial and transformed functions using graphing techniques to showcase the changes in slope and shape.
Vertical stretches affect the y-coordinates, leading to the function becoming 'taller' and the slope increasing.
In a vertical stretch, the y-coordinates of the function are multiplied by the stretch factor (e.g., multiplying by 2).
Graphing approach for vertical stretching: Observing how specific y-values (e.g., 1 becomes 2, -2 becomes -4) shift after transformation.
Visual demonstration of vertical stretch: The function grows taller, the y-intercept shifts, and the slope increases.
Key takeaway: Vertically stretching by a factor of two doubles the slope and y-intercept of the function.
Vertical stretches require multiplying the entire function by the stretch factor to get the new transformed equation.
Conclusion: Two methods to handle stretching transformations—graphing and using mathematical notation—both yield the same results.
Transcripts
in this video I'm going to talk about
stretching and compressing functions I'm
going to go over just a couple of
examples of how to horizontally and
vertically stretch a function okay so
little directions here uh let G of X be
the indicated transformation of and f of
x so this is down here is going to be
the transformation that we're going to
do okay and write the rule for G of X
okay so what we're going to do is we're
going to write the new equation for G
ofx okay so we're going to take this
function and we're going to horizontally
stretch it by a factor of two okay now
what you can well imagine though is that
when we when we horizontally stretch
something it's actually going to get
smaller the slope of it is going to go
down keep that in mind as we go through
this problem because when we check our
problem at the end to see if we did
things correctly that's what we're going
to base everything off of is that when
we horizontally stretch something if you
horizontally stretch actually the slope
is going to go down okay all right so
what what I'm going to do is I'm going
to show you with the notation and then
I'm gonna show you with the notation
first then graph it um just to show you
kind of two ways to do this all right so
what I'm going to do is I'm going to
take my function I'm going to change it
by now if I hor if I do something
horizontally to a function it only
affects the X portion of the function it
only affects the X so what I'm going to
do is I'm going to horizontally stretch
by a factor of two it's only going to
affect the X now here's my here's my
choice though I can either multiply
times two or divide by two that's
basically my two choices there so now I
got to think to myself is this function
going to get bigger or smaller now if I
horizontally stretch it it's act the
slope of it is going to get smaller so
over here on the graph if I have a
function that looks like this and then I
horizontally stretch it if I stretch
everything out it's actually going to
end up looking like this second one here
it's going to go from one to two Okay so
that right there gives you kind of an
idea of what's going to happen Okay
that's what happens when you
horizontally stretch the slope of your
function is actually going to go down
slope of the function is actually going
to go down okay I'll do a little bit
more exact drawing here in a minute or
graph here in a minute so what does that
mean the slope is going to go down which
means I have to divide by two okay so
I'm going to take the function and
divide by two or you could also say
multiply by 1/2 it does the same thing
okay so for my new function G of X for
the new one I'm going to take the old
one take the old one and multiply the X
portion of it times two or divide by two
or multiply by 1/2 same difference
so again I'm going to take the X portion
of it so three and I'm replace now
notice here x and then 12 x take the X
and replace it with a
1/2x take the X and replace it with a
1/2x that's basically what we're doing
okay so then my new function G of X my
new function G ofx is going to be so 3 /
2 that's just going to be 3es x - 2 3 x
- 2 okay so that's that's my new rule
that's the new function G of X that's
what it's going to look like now notice
the slope went from three to three
halves okay so the slope went down now
you can also think of three halves three
halves is uh one and one2 okay so it's
exactly half of three so you can see
there that it just went down okay so now
let's do that do that same thing let's
do it with graphing though okay show you
kind of a different way to do this I'm
going to take this function and graph it
-2 for my y intercept in a slope of
three 3 over 1 1 2 3 over 1 and 1 2 3
over
1 and then here is my
line there we go all right this is my f
function make sure you label so I know
which one is which and then now I'm
going to draw now what I'm going to do
is I'm going to take these points and if
I'm horizontally stretching by a factor
of two I take the x coordinates now
again remember horizontal so I take the
x coordinates and I multiply them times
two so the X coordinate here is two make
that a four okay x coordinate here is a
one make it a two see I'm just
multiplying times two doing it very
quickly this right here is an x
coordinate of zero okay so 0 * 2 is
still going to be zero so that point
stays right where it's at and then this
is an x coordinate of netive 1 * 2 would
be a negative -2 okay so these are my
new points for G of X these are my new
points for G of X wish I had a ruler on
this there we go all right so that's my
new function now notice that that right
there you can visually see we stretched
our function to get from F to G we
stretched everything out okay and now
what you can also see is um the Y
intercept is going to be -2 which that's
what it is and then my slope is going to
be 1 2 3 1 2 1 2 31 2 so this going be
2/3 positive 2/3 x so we did do that
correctly okay so there's just two
different ways to see it you can either
see it with the graphing or with the
notation there's two different ways to
do it all right now let's do another
example
stretching and compressing but this time
we're going to vertically stretch by a
factor of two so same same deal let G of
X be the indicated transformation of f
ofx write the rule for G ofx okay so
what we're going to do is we're going to
write the new equation but this time
what we're going to do is instead of
horizontally stretch we're actually
going to vertically stretch by a factor
of two okay now what I'm going to do
first is I'm actually going to do this
do this backwards from what I did last
time I'm going to graph it first figure
out what the equation is and then I'm
going to show it again do the problem
again but showing you how to do it with
a notation okay so I'm going to graph
this first so -2 for my Y intercept and
then 1 2 3 1 for my slope 1 two 3 1 for
my slope one two 3 1 for my slope and
there is my f function f function okay
now what I'm going to do is I'm going to
vertically stretch by a factor of two
which means I take now vertical stretch
vertical means I'm going to change the Y
coordinates so I'm going to take the Y
coordinates and multiply them times two
that's caus a little bit of trouble here
and you see Y in a second okay so right
here I have a y-coordinate of one so
take that times two is just going to be
two all right and then here I have a
y-coordinate of 1 2 3 4 so I have a
y-coordinate of four time 2 is going to
be eight which is going to be way up
here okay so I can't really graph that I
don't have a big enough graph to do that
okay so let's uh find something else
right here I have a y intercept of -2 -2
* 2 is4 so4 is down here all right very
good very good and then um now this
point here is at negative 1-23 4 5 it's
at5 * 2 is -10 which is going to be way
down here again I can't graph that point
but I have two points here that's that's
that's exactly what I need I need two
points to be able to write the equation
of a line all I need is two points okay
so right there that is going to be my G
function now notice we are vertically
stretching so notice everything is
getting taller everything's getting
taller the slope is getting bigger the Y
intercept is getting more negative you
can call that getting bigger uh little
things like that okay all right so then
my G of X function my G of X function
what's it going to be okay well I have
to find the slope in the Y intercept my
Y intercept is -4 so right there and
then my uh my slope I just got to count
that out 1 2 3 4 5 6 and one 6 over 1
which just reduces to six and then uh
it's a positive slop so I don't have to
change anything so there it is there is
the new rule for G ofx okay there's my
new function 6x - 4 okay now notice
compare that to the old one 3x - 2 all
we did was if we vertically stretched by
a factor of two you multiply the entire
function times two okay all right so
that's one way of seeing it with the
graphing now I'm going to show you again
with the notation so F ofx we're going
to change it by multiplying everything
times two if you vertically stretch
something if you vertically stretch
something you're going to multiply the
entire function times that number okay
or or divide by that number if you
vertically compress something because
everything's going to get smaller all
right so I my new G of X my new function
is going to be the old function is going
to be the old function except for I'm
just going to multiply times two so I
take the old function the old function
and just multiply times two okay take
the old function and multiply times two
and then this is the result that we
would get okay all right so there we go
there's two ways um two examples and two
ways to do both of examples you can
either graph or you can use the notation
to figure out what your new equation is
going to
be
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