How Is The Graph Of $y=\log(x)$ Transformed To Produce The Graph Of $y=\log(2x)+3$?A. It Is Stretched Horizontally By A Factor Of 2 And Translated Up 3 Units.B. It Is Compressed Horizontally By A Factor Of 2 And Translated Up 3

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Introduction

Graph transformations are a crucial concept in mathematics, particularly in algebra and calculus. They involve changing the shape or position of a graph to create a new graph. In this article, we will explore how the graph of $y=\log(x)$ is transformed to produce the graph of $y=\log(2x)+3$.

What are Graph Transformations?

Graph transformations are a way of changing the graph of a function to create a new graph. There are several types of graph transformations, including:

  • Horizontal Stretching: This involves stretching the graph horizontally by a factor of a certain value.
  • Horizontal Compressing: This involves compressing the graph horizontally by a factor of a certain value.
  • Vertical Stretching: This involves stretching the graph vertically by a factor of a certain value.
  • Vertical Compressing: This involves compressing the graph vertically by a factor of a certain value.
  • Translation: This involves moving the graph up or down by a certain value.

The Graph of $y=\log(x)$

The graph of $y=\log(x)$ is a logarithmic function that has a vertical asymptote at $x=0$. The graph is a curve that increases as $x$ increases.

The Graph of $y=\log(2x)+3$

The graph of $y=\log(2x)+3$ is a transformed version of the graph of $y=\log(x)$. To understand how the graph is transformed, we need to analyze the equation.

Analyzing the Equation

The equation $y=\log(2x)+3$ can be broken down into two parts:

  • \log(2x)$: This is a logarithmic function that has a vertical asymptote at $x=0$. However, the input of the logarithmic function is $2x$, which means that the graph is compressed horizontally by a factor of 2.

  • +3$: This is a vertical translation of the graph by 3 units.

Conclusion

Based on the analysis of the equation, we can conclude that the graph of $y=\log(2x)+3$ is transformed from the graph of $y=\log(x)$ by:

  • Horizontal Compressing: The graph is compressed horizontally by a factor of 2.
  • Vertical Translation: The graph is translated up by 3 units.

Therefore, the correct answer is:

B. It is compressed horizontally by a factor of 2 and translated up 3 units.

Example Problems

Here are some example problems to help you understand graph transformations:

  • Find the graph of $y=\log(3x)+2$.
  • Find the graph of $y=\log(x/2)+1$.
  • Find the graph of $y=\log(2x)+4$.

Solutions

  • The graph of $y=\log(3x)+2$ is transformed from the graph of $y=\log(x)$ by:
    • Horizontal Compressing: The graph is compressed horizontally by a factor of 3.
    • Vertical Translation: The graph is translated up by 2 units.
  • The graph of $y=\log(x/2)+1$ is transformed from the graph of $y=\log(x)$ by:
    • Horizontal Stretching: The graph is stretched horizontally by a factor of 2.
    • Vertical Translation: The graph is translated up by 1 unit.
  • The graph of $y=\log(2x)+4$ is transformed from the graph of $y=\log(x)$ by:
    • Horizontal Compressing: The graph is compressed horizontally by a factor of 2.
    • Vertical Translation: The graph is translated up by 4 units.

Conclusion

Q: What is the difference between horizontal stretching and horizontal compressing?

A: Horizontal stretching involves stretching the graph horizontally by a factor of a certain value, while horizontal compressing involves compressing the graph horizontally by a factor of a certain value.

Q: How do you determine the factor of horizontal stretching or compressing?

A: The factor of horizontal stretching or compressing is determined by the coefficient of the input variable in the logarithmic function. For example, in the equation $y=\log(2x)+3$, the coefficient of the input variable is 2, which means that the graph is compressed horizontally by a factor of 2.

Q: What is the effect of vertical translation on a graph?

A: Vertical translation involves moving the graph up or down by a certain value. This means that the graph is shifted vertically by the given value.

Q: How do you determine the value of vertical translation?

A: The value of vertical translation is determined by the constant term in the logarithmic function. For example, in the equation $y=\log(2x)+3$, the constant term is 3, which means that the graph is translated up by 3 units.

Q: Can you give an example of a graph that is stretched horizontally by a factor of 2?

A: Yes, the graph of $y=\log(2x)+2$ is an example of a graph that is stretched horizontally by a factor of 2.

Q: Can you give an example of a graph that is compressed horizontally by a factor of 3?

A: Yes, the graph of $y=\log(3x)+1$ is an example of a graph that is compressed horizontally by a factor of 3.

Q: How do you determine the type of transformation (stretching or compressing) when the coefficient of the input variable is greater than 1?

A: When the coefficient of the input variable is greater than 1, the graph is compressed horizontally by a factor of the coefficient. For example, in the equation $y=\log(4x)+2$, the coefficient of the input variable is 4, which means that the graph is compressed horizontally by a factor of 4.

Q: How do you determine the type of transformation (stretching or compressing) when the coefficient of the input variable is less than 1?

A: When the coefficient of the input variable is less than 1, the graph is stretched horizontally by a factor of the reciprocal of the coefficient. For example, in the equation $y=\log(x/2)+1$, the coefficient of the input variable is 1/2, which means that the graph is stretched horizontally by a factor of 2.

Q: Can you give an example of a graph that is translated up by 4 units?

A: Yes, the graph of $y=\log(x)+4$ is an example of a graph that is translated up by 4 units.

Q: Can you give an example of a graph that is translated down by 2 units?

A: Yes, the graph of $y=\log(x)-2$ is an example of a graph that is translated down by 2 units.

Conclusion

Graph transformations are a crucial concept in mathematics, particularly in algebra and calculus. By understanding how to transform graphs, you can create new graphs and solve problems more efficiently. In this article, we answered some common questions about graph transformations, including the difference between horizontal stretching and compressing, the effect of vertical translation, and how to determine the type of transformation.