Vertical Stretching and Compression

Oct 13, 2025 by magicalceylontours.com 149 views

Introduction

The exponential function $f(x) = 2^x$ undergoes two transformations to $g(x) = 3 \cdot 2^x + 5$ . In this article, we will explore how the graph of the function changes after these transformations. We will analyze the effects of the transformations on the graph, including vertical compression, vertical stretching, and horizontal shifting.

Vertical Stretching and Compression

The first transformation involves multiplying the function by a constant factor, $3$ . This is known as a vertical stretch. When a function is vertically stretched, its graph is stretched away from the x-axis. In this case, the graph of $g(x) = 3 \cdot 2^x + 5$ will be vertically stretched compared to the graph of $f(x) = 2^x$ .

On the other hand, if the function were multiplied by a constant factor less than 1, it would be vertically compressed. However, in this case, the function is multiplied by a constant factor greater than 1, so it is vertically stretched.

Horizontal Shifting

The second transformation involves adding a constant value, $5$ , to the function. This is known as a horizontal shift. When a function is shifted horizontally, its graph is shifted to the left or right. In this case, the graph of $g(x) = 3 \cdot 2^x + 5$ will be shifted to the right compared to the graph of $f(x) = 2^x$ .

To understand why the graph is shifted to the right, let's consider the equation $g(x) = 3 \cdot 2^x + 5$ . We can rewrite this equation as $g(x) = 3 \cdot 2^x + 5 = 3 \cdot 2^x + 5 - 5 + 5 = 3 \cdot 2^x + (5 - 5) + 5 = 3 \cdot 2^x + 5 - 5 + 5$ . This shows that the graph of $g(x)$ is shifted to the right by 5 units.

Conclusion

In conclusion, the graph of $g(x) = 3 \cdot 2^x + 5$ undergoes two transformations compared to the graph of $f(x) = 2^x$ . The first transformation involves a vertical stretch, while the second transformation involves a horizontal shift to the right. Therefore, the correct answers are:

B. It is vertically stretched.
C. It is shifted right.

Note that option A is incorrect because the graph is not vertically compressed. It is actually vertically stretched.

Key Takeaways

The graph of $g(x) = 3 \cdot 2^x + 5$ is vertically stretched compared to the graph of $f(x) = 2^x$ .
The graph of $g(x) = 3 \cdot 2^x + 5$ is shifted to the right compared to the graph of $f(x) = 2^x$ .
The vertical stretch is caused by multiplying the function by a constant factor greater than 1.
The horizontal shift is caused by adding a constant value to the function.

Exercises

What is the effect of multiplying the function $f(x) = 2^x$ by a constant factor less than 1?
What is the effect of adding a constant value to the function $f(x) = 2^x$ ?
How does the graph of $g(x) = 3 \cdot 2^x + 5$ change compared to the graph of $f(x) = 2^x$ ?

Answers

The function is vertically compressed.
The function is shifted to the left.
The graph of $g(x) = 3 \cdot 2^x + 5$ is vertically stretched and shifted to the right compared to the graph of $f(x) = 2^x$ .
Q&A: Understanding the Exponential Function Transformation ===========================================================

Introduction

In our previous article, we explored how the graph of the exponential function $f(x) = 2^x$ changes after undergoing two transformations to $g(x) = 3 \cdot 2^x + 5$ . We discussed the effects of vertical stretching and horizontal shifting on the graph. In this article, we will answer some frequently asked questions about the exponential function transformation.

Q: What is the effect of multiplying the function $f(x) = 2^x$ by a constant factor less than 1?

A: When you multiply the function $f(x) = 2^x$ by a constant factor less than 1, the graph of the function is vertically compressed. This means that the graph will be stretched towards the x-axis, making it narrower.

Q: What is the effect of adding a constant value to the function $f(x) = 2^x$ ?

A: When you add a constant value to the function $f(x) = 2^x$ , the graph of the function is shifted to the left. This means that the graph will be moved to the left by the amount of the constant value.

Q: How does the graph of $g(x) = 3 \cdot 2^x + 5$ change compared to the graph of $f(x) = 2^x$ ?

A: The graph of $g(x) = 3 \cdot 2^x + 5$ is vertically stretched and shifted to the right compared to the graph of $f(x) = 2^x$ . The vertical stretch is caused by multiplying the function by a constant factor greater than 1, and the horizontal shift is caused by adding a constant value to the function.

Q: What is the difference between a vertical stretch and a vertical compression?

A: A vertical stretch is when the graph of a function is stretched away from the x-axis, making it wider. A vertical compression is when the graph of a function is stretched towards the x-axis, making it narrower.

Q: What is the difference between a horizontal shift to the left and a horizontal shift to the right?

A: A horizontal shift to the left is when the graph of a function is moved to the left by a certain amount. A horizontal shift to the right is when the graph of a function is moved to the right by a certain amount.

Q: How can I determine whether a function is vertically stretched or compressed?

A: To determine whether a function is vertically stretched or compressed, you can look at the coefficient of the function. If the coefficient is greater than 1, the function is vertically stretched. If the coefficient is less than 1, the function is vertically compressed.

Q: How can I determine whether a function is shifted to the left or right?

A: To determine whether a function is shifted to the left or right, you can look at the constant term of the function. If the constant term is positive, the function is shifted to the right. If the constant term is negative, the function is shifted to the left.

Conclusion

In conclusion, the exponential function transformation is a powerful tool for understanding how graphs change under different transformations. By understanding the effects of vertical stretching and horizontal shifting, you can analyze and solve problems involving exponential functions. We hope this Q&A article has helped you understand the exponential function transformation better.

Key Takeaways

The graph of a function is vertically stretched when the coefficient is greater than 1.
The graph of a function is vertically compressed when the coefficient is less than 1.
The graph of a function is shifted to the right when the constant term is positive.
The graph of a function is shifted to the left when the constant term is negative.

Exercises

What is the effect of multiplying the function $f(x) = 2^x$ by a constant factor of 2?
What is the effect of adding a constant value of 3 to the function $f(x) = 2^x$ ?
How does the graph of $g(x) = 4 \cdot 2^x + 2$ change compared to the graph of $f(x) = 2^x$ ?

Answers

The function is vertically stretched.
The function is shifted to the right.
The graph of $g(x) = 4 \cdot 2^x + 2$ is vertically stretched and shifted to the right compared to the graph of $f(x) = 2^x$ .