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To graph the function [tex]\( y = 2 \log_2(-x + 7) - 3 \)[/tex], follow these detailed steps to identify at least five points with integer coordinates:
1. Understand the function:
[tex]\[ y = 2 \log_2(-x + 7) - 3 \][/tex]
This function is a logarithmic function with a base of 2 that has been horizontally translated and scaled.
2. Identify the domain:
The domain is determined by the argument of the logarithm being positive:
[tex]\[ -x + 7 > 0 \][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[ x < 7 \][/tex]
Hence, the domain of the function is [tex]\( x < 7 \)[/tex].
3. Set up the logarithmic expression:
Rewrite the function to focus on the inside of the log:
[tex]\[ y = 2 \log_2(7 - x) - 3 \][/tex]
This setup shows that the expression inside the log [tex]\( 7 - x \)[/tex] must be positive.
4. Find key points (calculate integer coordinates):
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 2 \log_2(7) - 3 \][/tex]
Using an approximate value for [tex]\( \log_2(7) \approx 2.807 \)[/tex]:
[tex]\[ y \approx 2 \times 2.807 - 3 \approx 5.614 - 3 \approx 2.614 \][/tex]
Rounding to the nearest integer:
[tex]\[ (0, 2.6) \rightarrow \text{(0, 3)} \][/tex]
- When [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 2 \log_2(6) - 3 \][/tex]
Using an approximate value for [tex]\( \log_2(6) \approx 2.585 \)[/tex]:
[tex]\[ y \approx 2 \times 2.585 - 3 \approx 5.17 - 3 \approx 2.17 \][/tex]
Rounding to the nearest integer:
[tex]\[ (1, 2.2) \rightarrow \text{(1, 2)} \][/tex]
- When [tex]\( x = 3 \)[/tex]:
[tex]\[ y = 2 \log_2(4) - 3 \][/tex]
Since [tex]\( \log_2(4) = 2 \)[/tex]:
[tex]\[ y = 2 \times 2 - 3 = 4 - 3 = 1 \][/tex]
[tex]\[ (3, 1) \][/tex]
- When [tex]\( x = 5 \)[/tex]:
[tex]\[ y = 2 \log_2(2) - 3 \][/tex]
Since [tex]\( \log_2(2) = 1 \)[/tex]:
[tex]\[ y = 2 \times 1 - 3 = 2 - 3 = -1 \][/tex]
[tex]\[ (5, -1) \][/tex]
- When [tex]\( x = 6 \)[/tex]:
[tex]\[ y = 2 \log_2(1) - 3 \][/tex]
Since [tex]\( \log_2(1) = 0 \)[/tex]:
[tex]\[ y = 2 \times 0 - 3 = 0 - 3 = -3 \][/tex]
[tex]\[ (6, -3) \][/tex]
5. Plot the points:
Plot the calculated points on the graph:
[tex]\[ (0, 3), (1, 2), (3, 1), (5, -1), (6, -3) \][/tex]
6. Connect the points:
Connect these points smoothly, keeping in mind the logarithmic nature of the function. The curve should approach negative infinity as [tex]\( x \)[/tex] approaches 7 from the left.
By plotting these five points with integer coordinates and connecting them appropriately, you should be able to graph [tex]\( y = 2 \log_2(-x + 7) - 3 \)[/tex].
1. Understand the function:
[tex]\[ y = 2 \log_2(-x + 7) - 3 \][/tex]
This function is a logarithmic function with a base of 2 that has been horizontally translated and scaled.
2. Identify the domain:
The domain is determined by the argument of the logarithm being positive:
[tex]\[ -x + 7 > 0 \][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[ x < 7 \][/tex]
Hence, the domain of the function is [tex]\( x < 7 \)[/tex].
3. Set up the logarithmic expression:
Rewrite the function to focus on the inside of the log:
[tex]\[ y = 2 \log_2(7 - x) - 3 \][/tex]
This setup shows that the expression inside the log [tex]\( 7 - x \)[/tex] must be positive.
4. Find key points (calculate integer coordinates):
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 2 \log_2(7) - 3 \][/tex]
Using an approximate value for [tex]\( \log_2(7) \approx 2.807 \)[/tex]:
[tex]\[ y \approx 2 \times 2.807 - 3 \approx 5.614 - 3 \approx 2.614 \][/tex]
Rounding to the nearest integer:
[tex]\[ (0, 2.6) \rightarrow \text{(0, 3)} \][/tex]
- When [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 2 \log_2(6) - 3 \][/tex]
Using an approximate value for [tex]\( \log_2(6) \approx 2.585 \)[/tex]:
[tex]\[ y \approx 2 \times 2.585 - 3 \approx 5.17 - 3 \approx 2.17 \][/tex]
Rounding to the nearest integer:
[tex]\[ (1, 2.2) \rightarrow \text{(1, 2)} \][/tex]
- When [tex]\( x = 3 \)[/tex]:
[tex]\[ y = 2 \log_2(4) - 3 \][/tex]
Since [tex]\( \log_2(4) = 2 \)[/tex]:
[tex]\[ y = 2 \times 2 - 3 = 4 - 3 = 1 \][/tex]
[tex]\[ (3, 1) \][/tex]
- When [tex]\( x = 5 \)[/tex]:
[tex]\[ y = 2 \log_2(2) - 3 \][/tex]
Since [tex]\( \log_2(2) = 1 \)[/tex]:
[tex]\[ y = 2 \times 1 - 3 = 2 - 3 = -1 \][/tex]
[tex]\[ (5, -1) \][/tex]
- When [tex]\( x = 6 \)[/tex]:
[tex]\[ y = 2 \log_2(1) - 3 \][/tex]
Since [tex]\( \log_2(1) = 0 \)[/tex]:
[tex]\[ y = 2 \times 0 - 3 = 0 - 3 = -3 \][/tex]
[tex]\[ (6, -3) \][/tex]
5. Plot the points:
Plot the calculated points on the graph:
[tex]\[ (0, 3), (1, 2), (3, 1), (5, -1), (6, -3) \][/tex]
6. Connect the points:
Connect these points smoothly, keeping in mind the logarithmic nature of the function. The curve should approach negative infinity as [tex]\( x \)[/tex] approaches 7 from the left.
By plotting these five points with integer coordinates and connecting them appropriately, you should be able to graph [tex]\( y = 2 \log_2(-x + 7) - 3 \)[/tex].
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