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.
