Answer :
Sure, let's graph the function [tex]\( f(x) = \sqrt{x - 4} - 3 \)[/tex].
### Step-by-Step Solution
1. Understand the Domain of the Function:
- The function involves a square root, and the expression under the square root, [tex]\( x - 4 \)[/tex], must be non-negative.
- Therefore, [tex]\( x - 4 \geq 0 \)[/tex].
- Solving this inequality, we obtain [tex]\( x \geq 4 \)[/tex].
- So, the domain of [tex]\( f(x) \)[/tex] is [tex]\( [4, \infty) \)[/tex].
2. Identify Key Points:
- Evaluate the function at a few key points within its domain:
- At [tex]\( x = 4 \)[/tex]:
[tex]\[ f(4) = \sqrt{4 - 4} - 3 = 0 - 3 = -3 \][/tex]
- At [tex]\( x = 5 \)[/tex]:
[tex]\[ f(5) = \sqrt{5 - 4} - 3 = 1 - 3 = -2 \][/tex]
- At [tex]\( x = 8 \)[/tex]:
[tex]\[ f(8) = \sqrt{8 - 4} - 3 = 2 - 3 = -1 \][/tex]
- At [tex]\( x = 13 \)[/tex]:
[tex]\[ f(13) = \sqrt{13 - 4} - 3 = 3 - 3 = 0 \][/tex]
- At [tex]\( x = 20 \)[/tex]:
[tex]\[ f(20) = \sqrt{20 - 4} - 3 = 4 - 3 = 1 \][/tex]
3. Basic Shape and Behavior of the Function:
- As [tex]\( x \)[/tex] increases from 4, [tex]\( \sqrt{x-4} \)[/tex] increases, and hence [tex]\( f(x) \)[/tex] increases as well.
- [tex]\( f(x) \)[/tex] is a transformation of the basic square root function [tex]\( g(x) = \sqrt{x} \)[/tex], shifted right by 4 units and down by 3 units.
4. Plotting the Graph:
- Axes: Draw the x-axis and y-axis.
- Plot the Points: Plot the points [tex]\((4, -3)\)[/tex], [tex]\((5, -2)\)[/tex], [tex]\((8, -1)\)[/tex], [tex]\((13, 0)\)[/tex], and [tex]\((20, 1)\)[/tex].
- Draw the Curve: Since [tex]\( f(x) \)[/tex] is increasing and non-linear (specifically, it’s a square root function), the curve should be a smooth increasing curve starting from [tex]\((4, -3)\)[/tex] and continuing to rise as [tex]\( x \)[/tex] increases.
5. Add Labels and Details:
- Label the x-axis as [tex]\( x \)[/tex] and the y-axis as [tex]\( f(x) \)[/tex].
- Add a title to the graph, such as "Graph of [tex]\( f(x) = \sqrt{x - 4} - 3 \)[/tex]".
- Label the significant points on the graph for clarity.
- Optionally, add grid lines for better readability.
### Resulting Graph
By following these steps, you would obtain a graph that starts at the point [tex]\((4, -3)\)[/tex] and increases as [tex]\( x \)[/tex] increases beyond 4. The graph resembles the right half of a sideways parabola, gradually rising without bounds as [tex]\( x \)[/tex] moves towards infinity.
You can use graphing tools or manually draw the curve carefully on graph paper to achieve the final visual representation of the function [tex]\( f(x) = \sqrt{x - 4} - 3 \)[/tex].
### Step-by-Step Solution
1. Understand the Domain of the Function:
- The function involves a square root, and the expression under the square root, [tex]\( x - 4 \)[/tex], must be non-negative.
- Therefore, [tex]\( x - 4 \geq 0 \)[/tex].
- Solving this inequality, we obtain [tex]\( x \geq 4 \)[/tex].
- So, the domain of [tex]\( f(x) \)[/tex] is [tex]\( [4, \infty) \)[/tex].
2. Identify Key Points:
- Evaluate the function at a few key points within its domain:
- At [tex]\( x = 4 \)[/tex]:
[tex]\[ f(4) = \sqrt{4 - 4} - 3 = 0 - 3 = -3 \][/tex]
- At [tex]\( x = 5 \)[/tex]:
[tex]\[ f(5) = \sqrt{5 - 4} - 3 = 1 - 3 = -2 \][/tex]
- At [tex]\( x = 8 \)[/tex]:
[tex]\[ f(8) = \sqrt{8 - 4} - 3 = 2 - 3 = -1 \][/tex]
- At [tex]\( x = 13 \)[/tex]:
[tex]\[ f(13) = \sqrt{13 - 4} - 3 = 3 - 3 = 0 \][/tex]
- At [tex]\( x = 20 \)[/tex]:
[tex]\[ f(20) = \sqrt{20 - 4} - 3 = 4 - 3 = 1 \][/tex]
3. Basic Shape and Behavior of the Function:
- As [tex]\( x \)[/tex] increases from 4, [tex]\( \sqrt{x-4} \)[/tex] increases, and hence [tex]\( f(x) \)[/tex] increases as well.
- [tex]\( f(x) \)[/tex] is a transformation of the basic square root function [tex]\( g(x) = \sqrt{x} \)[/tex], shifted right by 4 units and down by 3 units.
4. Plotting the Graph:
- Axes: Draw the x-axis and y-axis.
- Plot the Points: Plot the points [tex]\((4, -3)\)[/tex], [tex]\((5, -2)\)[/tex], [tex]\((8, -1)\)[/tex], [tex]\((13, 0)\)[/tex], and [tex]\((20, 1)\)[/tex].
- Draw the Curve: Since [tex]\( f(x) \)[/tex] is increasing and non-linear (specifically, it’s a square root function), the curve should be a smooth increasing curve starting from [tex]\((4, -3)\)[/tex] and continuing to rise as [tex]\( x \)[/tex] increases.
5. Add Labels and Details:
- Label the x-axis as [tex]\( x \)[/tex] and the y-axis as [tex]\( f(x) \)[/tex].
- Add a title to the graph, such as "Graph of [tex]\( f(x) = \sqrt{x - 4} - 3 \)[/tex]".
- Label the significant points on the graph for clarity.
- Optionally, add grid lines for better readability.
### Resulting Graph
By following these steps, you would obtain a graph that starts at the point [tex]\((4, -3)\)[/tex] and increases as [tex]\( x \)[/tex] increases beyond 4. The graph resembles the right half of a sideways parabola, gradually rising without bounds as [tex]\( x \)[/tex] moves towards infinity.
You can use graphing tools or manually draw the curve carefully on graph paper to achieve the final visual representation of the function [tex]\( f(x) = \sqrt{x - 4} - 3 \)[/tex].
