7D Example 8

Summary

The 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

00:00

📈 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.

05:03

🎨 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).

Keywords

💡Graph sketching

Graph sketching refers to the process of visually representing a mathematical function or equation on a coordinate plane. In the video, the speaker is explaining how to sketch the graph of the function y equals 10^x without using technology. The process involves understanding the behavior of the function, such as its asymptotes and zeros, and translating this into a visual representation on a graph.

💡Asymptotes

Asymptotes are lines that a function approaches but never actually reaches. In the context of the video, the speaker discusses the asymptotes of the sine and cosine functions, which are critical for sketching the graph of y = 10^x. The speaker mentions that the cosine function has zeros at π/2, 3π/2, etc., which become asymptotes for the tangent function, illustrating how these mathematical concepts are applied in graph sketching.

💡Period

The period of a function is the length of one complete cycle of the function's graph. In the video, the speaker explains that the period of the function y = 10^x is π, but when considering the function y = 10^(2x), the period is halved to π/2 due to the horizontal dilation. This change in period affects the frequency of oscillations in the graph, which is a key aspect of understanding the behavior of trigonometric functions.

💡Scale factor

A scale factor is a number that a function's graph is multiplied by, which affects the size of the graph. The speaker in the video discusses how the scale factor of the function y = 10^(2x) is 1/2, which results in the graph being reduced by half. This concept is crucial for understanding how transformations of functions, such as horizontal dilations, affect the appearance of the graph.

💡Horizontal dilation

Horizontal dilation is a transformation that stretches or compresses a graph horizontally. The video script mentions that the period of y = 10^(2x) is π/2, indicating a horizontal dilation by a factor of 1/2. This means that the graph is stretched horizontally, leading to more frequent oscillations within the same interval, which is a key concept in the transformation of trigonometric functions.

💡X-intercepts

X-intercepts are the points where a graph crosses the x-axis. In the video, the speaker discusses finding the x-intercepts of the function y = 10^x, which are essential for accurately sketching the graph. The x-intercepts for the function y = 10^(2x) are calculated as π/2, -π/2, and so on, which are then used to mark the points where the graph intersects the x-axis.

💡Tangent function

The tangent function, often abbreviated as tan(x), is a trigonometric function that represents the ratio of the sine to the cosine of an angle. In the video, the speaker is sketching the graph of y = 10^x and then discusses how to modify it to sketch y = 10^(2x), which involves understanding the tangent function's behavior and its relationship to the sine and cosine functions.

💡Transformation

Transformation in the context of graph sketching refers to the process of modifying a graph by applying various operations such as translations, dilations, and reflections. The video script describes how the graph of y = 10^x is transformed into y = 10^(2x) by applying a horizontal dilation and a change in the scale factor, which alters the graph's appearance and behavior.

💡Zeroes

Zeroes of a function are the x-values for which the function's output is zero. In the video, the speaker identifies the zeroes of the sine and cosine functions, which are crucial for understanding where the graph of y = 10^x will intersect the x-axis or have asymptotes. The zeroes are used to determine the x-intercepts and to sketch the correct behavior of the graph.

💡Trigonometric functions

Trigonometric functions, such as sine, cosine, and tangent, are fundamental in mathematics and are used to describe the relationships between the angles and sides of triangles. In the video, the speaker uses the properties of these functions to sketch the graph of y = 10^x and its transformations, demonstrating how these functions are integral to understanding the behavior of more complex functions.

💡Dilation

Dilation, in the context of graph sketching, refers to the process of enlarging or reducing the size of a graph. The video script mentions a horizontal dilation of the graph of y = 10^x when transforming it into y = 10^(2x), which results in a change in the period and the scale of the graph. This concept is essential for understanding how the graph's appearance changes with transformations.

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