Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Connect with a community of experts ready to provide precise solutions to your questions on our user-friendly Q&A platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
Certainly! Let's analyze the function [tex]\( f(x) = -\sqrt{x} - 4 \)[/tex] and understand how it behaves. Here are the steps to determine the correct graph for this function:
1. Understand the Function:
- The function [tex]\( f(x) = -\sqrt{x} - 4 \)[/tex] is a transformation of the basic square root function.
- The square root function [tex]\( \sqrt{x} \)[/tex] typically starts at the origin (0,0) and increases as [tex]\( x \)[/tex] increases.
- Here, we have a negative sign in front of the square root, so this inverts the square root function vertically (it will go downwards as [tex]\( x \)[/tex] increases).
2. Transformation Details:
- The term [tex]\( -\sqrt{x} \)[/tex] means the curve will start at the origin and go downwards as [tex]\( x \)[/tex] increases.
- The additional [tex]\(-4\)[/tex] shifts the entire graph 4 units downwards.
3. Key Points Calculation:
- Let's calculate a few points to understand how the graph will look like:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -\sqrt{0} - 4 = -4 \][/tex]
So, the point (0, -4) is on the graph.
- When [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = -\sqrt{1} - 4 = -1 - 4 = -5 \][/tex]
So, the point (1, -5) is on the graph.
- When [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = -\sqrt{2} - 4 \approx -1.414 - 4 = -5.414 \][/tex]
So, the point (2, -5.414) is on the graph.
- When [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = -\sqrt{3} - 4 \approx -1.732 - 4 = -5.732 \][/tex]
So, the point (3, -5.732) is on the graph.
- When [tex]\( x = 4 \)[/tex]:
[tex]\[ f(4) = -\sqrt{4} - 4 = -2 - 4 = -6 \][/tex]
So, the point (4, -6) is on the graph.
- When [tex]\( x = 5 \)[/tex]:
[tex]\[ f(5) = -\sqrt{5} - 4 \approx -2.236 - 4 = -6.236 \][/tex]
So, the point (5, -6.236) is on the graph.
- When [tex]\( x = 6 \)[/tex]:
[tex]\[ f(6) = -\sqrt{6} - 4 \approx -2.449 - 4 = -6.449 \][/tex]
So, the point (6, -6.449) is on the graph.
- When [tex]\( x = 7 \)[/tex]:
[tex]\[ f(7) = -\sqrt{7} - 4 \approx -2.646 - 4 = -6.646 \][/tex]
So, the point (7, -6.646) is on the graph.
- When [tex]\( x = 8 \)[/tex]:
[tex]\[ f(8) = -\sqrt{8} - 4 \approx -2.828 - 4 = -6.828 \][/tex]
So, the point (8, -6.828) is on the graph.
- When [tex]\( x = 9 \)[/tex]:
[tex]\[ f(9) = -\sqrt{9} - 4 = -3 - 4 = -7 \][/tex]
So, the point (9, -7) is on the graph.
- When [tex]\( x = 10 \)[/tex]:
[tex]\[ f(10) = -\sqrt{10} - 4 \approx -3.162 - 4 = -7.162 \][/tex]
So, the point (10, -7.162) is on the graph.
4. Graph Behavior:
- The graph starts at (0, -4) and moves downward as [tex]\( x \)[/tex] increases, but the rate of descent decreases as [tex]\( x \)[/tex] increases due to the square root factor.
5. Conclusion:
- The function [tex]\( f(x) = -\sqrt{x} - 4 \)[/tex] describes a curve starting at (0, -4) and gradually decreasing more slowly as [tex]\( x \)[/tex] increases.
To decide which graph among the given options (A, B, C, D) correctly represents this behavior, you would look for a graph starting at (0, -4) and then decreasing in a concave manner.
(As the provided graphs are not shown here, I cannot select a graph directly. However, now you know what behavior to look for: start at (0, -4), decrease quickly at first, and then more slowly as [tex]\( x \)[/tex] increases.)
1. Understand the Function:
- The function [tex]\( f(x) = -\sqrt{x} - 4 \)[/tex] is a transformation of the basic square root function.
- The square root function [tex]\( \sqrt{x} \)[/tex] typically starts at the origin (0,0) and increases as [tex]\( x \)[/tex] increases.
- Here, we have a negative sign in front of the square root, so this inverts the square root function vertically (it will go downwards as [tex]\( x \)[/tex] increases).
2. Transformation Details:
- The term [tex]\( -\sqrt{x} \)[/tex] means the curve will start at the origin and go downwards as [tex]\( x \)[/tex] increases.
- The additional [tex]\(-4\)[/tex] shifts the entire graph 4 units downwards.
3. Key Points Calculation:
- Let's calculate a few points to understand how the graph will look like:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -\sqrt{0} - 4 = -4 \][/tex]
So, the point (0, -4) is on the graph.
- When [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = -\sqrt{1} - 4 = -1 - 4 = -5 \][/tex]
So, the point (1, -5) is on the graph.
- When [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = -\sqrt{2} - 4 \approx -1.414 - 4 = -5.414 \][/tex]
So, the point (2, -5.414) is on the graph.
- When [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = -\sqrt{3} - 4 \approx -1.732 - 4 = -5.732 \][/tex]
So, the point (3, -5.732) is on the graph.
- When [tex]\( x = 4 \)[/tex]:
[tex]\[ f(4) = -\sqrt{4} - 4 = -2 - 4 = -6 \][/tex]
So, the point (4, -6) is on the graph.
- When [tex]\( x = 5 \)[/tex]:
[tex]\[ f(5) = -\sqrt{5} - 4 \approx -2.236 - 4 = -6.236 \][/tex]
So, the point (5, -6.236) is on the graph.
- When [tex]\( x = 6 \)[/tex]:
[tex]\[ f(6) = -\sqrt{6} - 4 \approx -2.449 - 4 = -6.449 \][/tex]
So, the point (6, -6.449) is on the graph.
- When [tex]\( x = 7 \)[/tex]:
[tex]\[ f(7) = -\sqrt{7} - 4 \approx -2.646 - 4 = -6.646 \][/tex]
So, the point (7, -6.646) is on the graph.
- When [tex]\( x = 8 \)[/tex]:
[tex]\[ f(8) = -\sqrt{8} - 4 \approx -2.828 - 4 = -6.828 \][/tex]
So, the point (8, -6.828) is on the graph.
- When [tex]\( x = 9 \)[/tex]:
[tex]\[ f(9) = -\sqrt{9} - 4 = -3 - 4 = -7 \][/tex]
So, the point (9, -7) is on the graph.
- When [tex]\( x = 10 \)[/tex]:
[tex]\[ f(10) = -\sqrt{10} - 4 \approx -3.162 - 4 = -7.162 \][/tex]
So, the point (10, -7.162) is on the graph.
4. Graph Behavior:
- The graph starts at (0, -4) and moves downward as [tex]\( x \)[/tex] increases, but the rate of descent decreases as [tex]\( x \)[/tex] increases due to the square root factor.
5. Conclusion:
- The function [tex]\( f(x) = -\sqrt{x} - 4 \)[/tex] describes a curve starting at (0, -4) and gradually decreasing more slowly as [tex]\( x \)[/tex] increases.
To decide which graph among the given options (A, B, C, D) correctly represents this behavior, you would look for a graph starting at (0, -4) and then decreasing in a concave manner.
(As the provided graphs are not shown here, I cannot select a graph directly. However, now you know what behavior to look for: start at (0, -4), decrease quickly at first, and then more slowly as [tex]\( x \)[/tex] increases.)
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
