Join the growing community of curious minds on IDNLearn.com and get the answers you need. Our Q&A platform offers detailed and trustworthy answers to ensure you have the information you need.
Sagot :
To graph the function [tex]\( y = -\log_2(x + 7) + 5 \)[/tex] and find some significant points, we will follow these steps:
1. Understand the Domain: The function includes a logarithmic term [tex]\( \log_2(x + 7) \)[/tex], where [tex]\( \log_2 \)[/tex] denotes the logarithm base 2. For this function to be defined, the argument of the logarithm must be positive, i.e., [tex]\( x + 7 > 0 \Rightarrow x > -7 \)[/tex]. Thus, the domain of [tex]\( y \)[/tex] is [tex]\( x > -7 \)[/tex].
2. Identify Key Points:
- For several chosen values of [tex]\( x \)[/tex], we will calculate the corresponding [tex]\( y \)[/tex]-values.
- Key points within the domain might include [tex]\( x = -6, -5, 0, 1, \)[/tex] and [tex]\( 3 \)[/tex].
3. Calculate Key Points:
- For [tex]\( x = -6 \)[/tex]:
[tex]\[ y = -\log_2(-6 + 7) + 5 = -\log_2(1) + 5 = -0 + 5 = 5 \][/tex]
Hence, the point is [tex]\((-6, 5)\)[/tex].
- For [tex]\( x = -5 \)[/tex]:
[tex]\[ y = -\log_2(-5 + 7) + 5 = -\log_2(2) + 5 = -1 + 5 = 4 \][/tex]
Hence, the point is [tex]\((-5, 4)\)[/tex].
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -\log_2(0 + 7) + 5 \approx -\log_2(7) + 5 \approx 2.193 \][/tex]
Hence, the point is [tex]\((0, 2.193)\)[/tex].
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ y = -\log_2(1 + 7) + 5 \approx -\log_2(8) + 5 = -3 + 5 = 2 \][/tex]
Hence, the point is [tex]\((1, 2)\)[/tex].
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ y = -\log_2(3 + 7) + 5 \approx -\log_2(10) + 5 \approx 1.678 \][/tex]
Hence, the point is [tex]\((3, 1.678)\)[/tex].
4. Generate Additional Points for Graph: Using technology, we can generate additional points to create a smooth graph. For instance, we can evaluate [tex]\( y \)[/tex] for [tex]\( x \)[/tex] values ranging from just above -7 (e.g., -6.9) to some larger positive values (e.g., 10).
5. Create Graph:
- Plot the key points calculated: [tex]\((-6, 5)\)[/tex], [tex]\((-5, 4)\)[/tex], [tex]\((0, 2.193)\)[/tex], [tex]\((1, 2)\)[/tex], and [tex]\((3, 1.678)\)[/tex].
- Plot the additional points we generated to ensure a smooth and accurate graph.
6. Visualize the Function:
- By placing all the points on a graph with the [tex]\(x\)[/tex]-axis from just above [tex]\(-7\)[/tex] to 10 and the [tex]\(y\)[/tex]-axis from 0 to 6, we can connect these points smoothly to get an accurate representation of the function [tex]\( y = -\log_2(x + 7) + 5 \)[/tex].
Thus, by carefully calculating and plotting these key points carefully along with additional values, we form the graph of [tex]\( y = -\log_2(x + 7) + 5 \)[/tex]. This function shows a decreasing trend, approaching [tex]\( \infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\(\ -7\)[/tex] (from the right), and gradually decreasing values as [tex]\( x \)[/tex] increases.
1. Understand the Domain: The function includes a logarithmic term [tex]\( \log_2(x + 7) \)[/tex], where [tex]\( \log_2 \)[/tex] denotes the logarithm base 2. For this function to be defined, the argument of the logarithm must be positive, i.e., [tex]\( x + 7 > 0 \Rightarrow x > -7 \)[/tex]. Thus, the domain of [tex]\( y \)[/tex] is [tex]\( x > -7 \)[/tex].
2. Identify Key Points:
- For several chosen values of [tex]\( x \)[/tex], we will calculate the corresponding [tex]\( y \)[/tex]-values.
- Key points within the domain might include [tex]\( x = -6, -5, 0, 1, \)[/tex] and [tex]\( 3 \)[/tex].
3. Calculate Key Points:
- For [tex]\( x = -6 \)[/tex]:
[tex]\[ y = -\log_2(-6 + 7) + 5 = -\log_2(1) + 5 = -0 + 5 = 5 \][/tex]
Hence, the point is [tex]\((-6, 5)\)[/tex].
- For [tex]\( x = -5 \)[/tex]:
[tex]\[ y = -\log_2(-5 + 7) + 5 = -\log_2(2) + 5 = -1 + 5 = 4 \][/tex]
Hence, the point is [tex]\((-5, 4)\)[/tex].
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -\log_2(0 + 7) + 5 \approx -\log_2(7) + 5 \approx 2.193 \][/tex]
Hence, the point is [tex]\((0, 2.193)\)[/tex].
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ y = -\log_2(1 + 7) + 5 \approx -\log_2(8) + 5 = -3 + 5 = 2 \][/tex]
Hence, the point is [tex]\((1, 2)\)[/tex].
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ y = -\log_2(3 + 7) + 5 \approx -\log_2(10) + 5 \approx 1.678 \][/tex]
Hence, the point is [tex]\((3, 1.678)\)[/tex].
4. Generate Additional Points for Graph: Using technology, we can generate additional points to create a smooth graph. For instance, we can evaluate [tex]\( y \)[/tex] for [tex]\( x \)[/tex] values ranging from just above -7 (e.g., -6.9) to some larger positive values (e.g., 10).
5. Create Graph:
- Plot the key points calculated: [tex]\((-6, 5)\)[/tex], [tex]\((-5, 4)\)[/tex], [tex]\((0, 2.193)\)[/tex], [tex]\((1, 2)\)[/tex], and [tex]\((3, 1.678)\)[/tex].
- Plot the additional points we generated to ensure a smooth and accurate graph.
6. Visualize the Function:
- By placing all the points on a graph with the [tex]\(x\)[/tex]-axis from just above [tex]\(-7\)[/tex] to 10 and the [tex]\(y\)[/tex]-axis from 0 to 6, we can connect these points smoothly to get an accurate representation of the function [tex]\( y = -\log_2(x + 7) + 5 \)[/tex].
Thus, by carefully calculating and plotting these key points carefully along with additional values, we form the graph of [tex]\( y = -\log_2(x + 7) + 5 \)[/tex]. This function shows a decreasing trend, approaching [tex]\( \infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\(\ -7\)[/tex] (from the right), and gradually decreasing values as [tex]\( x \)[/tex] increases.
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com provides the best answers to your questions. Thank you for visiting, and come back soon for more helpful information.
