7D Example 8
Summary
TLDRThe video script explains the process of sketching the graph of the function y equals 10 to the x without using technology. It starts by discussing the sine function, identifying its zeros and asymptotes. The tutorial then moves on to graph y equals 10 to 2x, explaining the effects of horizontal dilation and period adjustment. The instructor sketches the graph, adjusting for the new period and scale factor, and marking the x-intercepts. The final graph shows more oscillations due to the halved period, providing a clear visual of the function's behavior.
Takeaways
- 📐 The video script is a tutorial on sketching the graph of y = tan(x) without using technology.
- 📈 The instructor emphasizes the importance of understanding the basic properties of the sine and cosine functions for graphing.
- 🔍 The graph is sketched over the interval from -π to π, focusing on the significant points within this range.
- 📍 The zeros of the sine function and the zeros of the cosine function are identified to establish the asymptotes of the tangent function.
- 📉 The original function y = tan(x) is graphed, showing its characteristic oscillations and asymptotes at ±π/2.
- 🔄 The transformation to y = tan(2x) involves a horizontal dilation, which affects the period and the position of the asymptotes.
- 🔢 The period of the tangent function is halved when the argument of the tangent function is multiplied by 2, changing from π to π/2.
- 📉 The scale factor for the transformation is 1/2, which causes the graph to compress horizontally.
- 📌 New asymptotes are calculated as ±π/4, resulting from the horizontal dilation of the original asymptotes.
- 🖊️ The final graph of y = tan(2x) is sketched, showing more frequent oscillations due to the reduced period and the new positions of the asymptotes.
Q & A
What is the main topic discussed in the video script?
-The main topic discussed in the video script is the process of sketching the graph of the function y equals 10 to the power of x without using technology.
What is the first step the instructor takes in sketching the graph?
-The first step the instructor takes is to determine the range of x values, which is from -π to π.
Why does the instructor choose to focus on the window from -π to π?
-The instructor focuses on the window from -π to π because it is the standard range for sketching trigonometric functions like sine and cosine, which are related to the tangent function being discussed.
What are the zeros of the sine function as mentioned in the script?
-The zeros of the sine function mentioned in the script are at x = π and x = 2π.
What are the asymptotes for the tangent function as discussed in the script?
-The asymptotes for the tangent function are at x = π/2 and x = -π/2.
How does the instructor modify the graph of y = 10^x to get y = 10^(2x)?
-The instructor modifies the graph of y = 10^x to get y = 10^(2x) by applying a horizontal dilation, reducing the period to π/2 and scaling the graph by a factor of 1/2.
What is the new period of the function y = 10^(2x) after the modification?
-The new period of the function y = 10^(2x) after the modification is π/2.
What are the new asymptotes for the function y = 10^(2x) after the modification?
-The new asymptotes for the function y = 10^(2x) after the modification are at x = π/4 and x = -π/4.
How does the instructor indicate x-intercepts on the graph?
-The instructor indicates x-intercepts on the graph by marking them with black dots at the points where the function crosses the x-axis.
What is the final appearance of the graph for y = 10^(2x) according to the script?
-The final appearance of the graph for y = 10^(2x) is a series of oscillations between the new asymptotes, with the period halved and more oscillations visible within the window from -π to π.
Outlines
📈 Sketching the Graph of y = 10sin(2x)
The paragraph describes the process of sketching the graph of the function y = 10sin(2x) without the use of technology. The speaker begins by indicating the need to draw the graph within the range of -π to π, highlighting the importance of drawing the sine function and its zeros at multiples of π and 2π. The speaker then identifies the asymptotes of the cosine function at π/2 and 3π/2, which will be crucial for the tangent function. The original function is sketched, followed by the transformation to the tangent function, y = 10tan(2x). The speaker notes the effects of the transformation, including a horizontal dilation by a factor of π/2 and a scale factor of 1/2, which results in more frequent oscillations. The asymptotes are adjusted accordingly, and the x-intercepts are identified and marked. The final graph is described as having more oscillations due to the halved period, with the sine function's x-intercepts remaining unchanged within the specified window.
🎨 Final Graph of y = 10tan(2x)
The second paragraph concludes the process by presenting the final graph of y = 10tan(2x). The speaker confirms the appearance of the graph, which includes the adjusted asymptotes and x-intercepts as discussed in the previous paragraph. The graph is described as complete, with all the necessary elements such as the period, scale factor, and asymptotes properly depicted. The speaker seems satisfied with the final result, indicating that the graph accurately represents the function y = 10tan(2x).
Mindmap
Keywords
💡Graph sketching
💡Asymptotes
💡Period
💡Scale factor
💡Horizontal dilation
💡X-intercepts
💡Tangent function
💡Transformation
💡Zeroes
💡Trigonometric functions
💡Dilation
Highlights
Introduction to sketching the graph of y equals 10^x without using technology.
Explanation of the need to draw the graph from -π to π.
Preference for placing the origin in the middle when sketching parameter graphs.
Identification of key points such as 3π/2 and 2π for graph sketching.
Explanation of the sine function zeros at multiples of π.
Description of the cosine function zeros at π/2 intervals.
Mention of asymptotes for the sine function at π/2 and multiples.
Initial sketch of the sine function graph with zeros and asymptotes.
Transformation to the tangent function by manipulating the sine function.
Note on the scale factor and period changes when transforming to tangent function.
Horizontal dilation of the graph with a period of π/2.
Scale factor adjustment to one over two, reducing the graph by half.
Reduction of the asymptotes to π/4 intervals due to the transformation.
Sketching the updated graph with reduced asymptotes and period.
Identification of x-intercepts for the tangent function graph.
Use of black dotted lines to indicate the new asymptotes.
Use of black dots to mark x-intercepts on the graph.
Final graph presentation showing more oscillations due to the halved period.
Completion of the graph sketch for y equals tan(x).
Transcripts
go
so again using without using technology
sketch the graph of y equals 10 to X
something like this you could definitely
be
hind calculated section
okay first I'm not going to explain in
detail how to draw the um 10 Theta
function hopefully you have listened to
the previous example in lots of videos
for you to be able to know just
realizing that I need to draw a minus pi
to Pi so that's why I do is kind of in
the middle I always like to do
parameter
all right
three pi over two
two Pi I don't really need that much I
only need up to Pi but anyway I've got
negative 2 oh negative pi over 2 and
then Pi negative pi let me rewrite this
because it looks very messy negative pi
over 2.
okay so we're really only looking at
that window
um so let's give it a go my Japan X I
have my sine function so my sine X
is zero when X is
um all the answers pi and 2 pi
all right then my zeros
and then here as well and then
um my cos x is zero one
X is pi over two three four over two
Etc
so those are my asymptotes so pi over
two
and minus five
all right so this is my original
function let me give it a draw for you
this one so it looks something like this
[Music]
file should have done as well
so it looks something like this
here we go so that's what my graph looks
like and then what I should have is tan
2x so let me write y equals 10x my y
equals
10 2x have to be manipulated a little
bit let's see what oh let's see what the
sorry to say so if the B is inside and
it's greater than zero it has a scale
factor of one over B the graph has a
period pi over B horizontal dilation so
those are the notes that I want to use
so I'll do it quickly because I'm
running out of charge
horizontal dilation
period is pi over two now
um and then the scale factor is
one over two so it's going to reduce by
half and my pi over 2 is my period so at
the moment my period is pi over two to
pi over two so it's a whole Pi but um
it's going to reduce it it's going to be
more oscillations
so let's give it a go
everything is going to basically Reduce
by pi over 2.
so my original function is going to be
the same but my asymptote is going to
reduce
so maybe I'll use black dotted line
um so it's pi over two at the moment if
I do pi over two so asymptote
to divide two it's going to be pi over
four my other one's going to be minus pi
over two divide two so it's going to be
minus pi over four Etc so these are like
my new
because sometimes ah and that means that
every pi over 4 there's going to be an
asymptote because
um
how it works there's another asymptote
and this is yeah
okay that graph is looking a bit messy
but we'll keep going and we'll do the X
intercepts so my current x-intercept is
zero which will stay
Pi divide by 2 so that'll be part of two
and then minus pi divided by 2 3 minus
pi over 2 and then it will extend
further as well I might use
um so I'll just use black dots to show
you that so pi over 2 is over here
pi over minus pi over 2 is over here and
then obviously the other dots will come
in more that way as well and that one
stays so let me see if I can use a
different color maybe green light green
to show that so this is what my graph
would look like now
stays in between those windows as you
can see it my period halved so we are
seeing more oscillations
so it's something like that
okay and then
it looks like this
there we go so there we go y equals ten
to X done
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