Answer :
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.)
