Get the answers you've been looking for with the help of IDNLearn.com's expert community. Get accurate answers to your questions from our community of experts who are always ready to provide timely and relevant solutions.
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.
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Your questions are important to us at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.
