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To understand how the graph of [tex]\( g(x) = x - 9 \)[/tex] can be obtained from one of the basic graphs, let's break it down step by step.
1. Identify the Basic Graph:
The function [tex]\( g(x) = x - 9 \)[/tex] can be related to the basic linear function [tex]\( f(x) = x \)[/tex], which is a straight line passing through the origin with a slope of 1.
2. Transformation:
The equation [tex]\( g(x) = x - 9 \)[/tex] indicates a vertical shift of the graph of [tex]\( f(x) = x \)[/tex].
- The term "-9" means that every point on the graph of the basic function [tex]\( f(x) \)[/tex] is shifted downward by 9 units.
### Step-by-step Process to Transform [tex]\( f(x) = x \)[/tex] to [tex]\( g(x) = x - 9 \)[/tex]:
1. Start with the basic graph [tex]\( f(x) = x \)[/tex]:
This graph is a straight line that passes through the origin (0, 0) and has a slope of 1. For example, it includes points like (0, 0), (1, 1), (2, 2), etc.
2. Apply the vertical shift:
- Subtract 9 from each y-coordinate of the points on the graph of [tex]\( f(x) \)[/tex].
- For example, the point (0, 0) on [tex]\( f(x) \)[/tex] will be transformed to (0, 0 - 9), which is (0, -9).
- Similarly, (1, 1) on [tex]\( f(x) \)[/tex] will become (1, 1 - 9), which is (1, -8), and (2, 2) will become (2, 2 - 9) or (2, -7).
### Graphing [tex]\( g(x) = x - 9 \)[/tex]:
1. Plot key points:
- You can plot several points based on the transformed y-values. For instance:
- (0, -9)
- (1, -8)
- (2, -7)
- (-1, -10)
- (-2, -11)
2. Draw the line:
- Draw a straight line passing through these points.
- The line should maintain the same slope (1), meaning it should rise one unit vertically for every one unit it moves horizontally, just as the original line [tex]\( f(x) = x \)[/tex] does, but it will be shifted 9 units down.
### Graph Overview:
- The line will look exactly like the line [tex]\( f(x) = x \)[/tex], but all points on the line will be lower by 9 units.
- This graph will cross the y-axis at -9 (rather than 0).
Here's a simple illustration to help visualize this transformation:
1. f(x) = x (Basic Graph):
```
|
2 |
1 |
0 ----------------
-1 |
-2 |
-2 -1 0 1 2
```
2. g(x) = x - 9 (Transformed Graph):
```
|
9 - |
8 - |
7 - |
|
0 ----------------
-1 -|
-2 -|
- -8 -7 -6 ... 0 ... 7 8 ...
```
In this illustration, the second graph is vertically shifted downward by [tex]\( 9 \)[/tex] units. Every y-value from [tex]\( f(x) = x \)[/tex] is decreased by 9 to create [tex]\( g(x) = x - 9 \)[/tex].
1. Identify the Basic Graph:
The function [tex]\( g(x) = x - 9 \)[/tex] can be related to the basic linear function [tex]\( f(x) = x \)[/tex], which is a straight line passing through the origin with a slope of 1.
2. Transformation:
The equation [tex]\( g(x) = x - 9 \)[/tex] indicates a vertical shift of the graph of [tex]\( f(x) = x \)[/tex].
- The term "-9" means that every point on the graph of the basic function [tex]\( f(x) \)[/tex] is shifted downward by 9 units.
### Step-by-step Process to Transform [tex]\( f(x) = x \)[/tex] to [tex]\( g(x) = x - 9 \)[/tex]:
1. Start with the basic graph [tex]\( f(x) = x \)[/tex]:
This graph is a straight line that passes through the origin (0, 0) and has a slope of 1. For example, it includes points like (0, 0), (1, 1), (2, 2), etc.
2. Apply the vertical shift:
- Subtract 9 from each y-coordinate of the points on the graph of [tex]\( f(x) \)[/tex].
- For example, the point (0, 0) on [tex]\( f(x) \)[/tex] will be transformed to (0, 0 - 9), which is (0, -9).
- Similarly, (1, 1) on [tex]\( f(x) \)[/tex] will become (1, 1 - 9), which is (1, -8), and (2, 2) will become (2, 2 - 9) or (2, -7).
### Graphing [tex]\( g(x) = x - 9 \)[/tex]:
1. Plot key points:
- You can plot several points based on the transformed y-values. For instance:
- (0, -9)
- (1, -8)
- (2, -7)
- (-1, -10)
- (-2, -11)
2. Draw the line:
- Draw a straight line passing through these points.
- The line should maintain the same slope (1), meaning it should rise one unit vertically for every one unit it moves horizontally, just as the original line [tex]\( f(x) = x \)[/tex] does, but it will be shifted 9 units down.
### Graph Overview:
- The line will look exactly like the line [tex]\( f(x) = x \)[/tex], but all points on the line will be lower by 9 units.
- This graph will cross the y-axis at -9 (rather than 0).
Here's a simple illustration to help visualize this transformation:
1. f(x) = x (Basic Graph):
```
|
2 |
1 |
0 ----------------
-1 |
-2 |
-2 -1 0 1 2
```
2. g(x) = x - 9 (Transformed Graph):
```
|
9 - |
8 - |
7 - |
|
0 ----------------
-1 -|
-2 -|
- -8 -7 -6 ... 0 ... 7 8 ...
```
In this illustration, the second graph is vertically shifted downward by [tex]\( 9 \)[/tex] units. Every y-value from [tex]\( f(x) = x \)[/tex] is decreased by 9 to create [tex]\( g(x) = x - 9 \)[/tex].
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