7D Example 8

Miss Kim
20 May 202305:17

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

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Mindmap

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Keywords

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Highlights

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Transcripts

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Ähnliche Tags
Graph SketchingFunction TransformationAsymptotesMath TutorialTrigonometryEducational ContentMathematicsVisual LearningX-interceptsPeriod Manipulation
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