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Sagot :
To determine which graph corresponds to the function [tex]\(\frac{1}{3}f(x)\)[/tex], let's first analyze the function [tex]\(f(x)\)[/tex] and how [tex]\(\frac{1}{3} f(x)\)[/tex] behaves.
### Analyzing [tex]\(f(x)\)[/tex]:
Given [tex]\(f(x) = x - 3\)[/tex], this function represents a straight line with a slope of 1 and a y-intercept of [tex]\(-3\)[/tex]. The equation implies that:
- For every increase of 1 in [tex]\(x\)[/tex], [tex]\(f(x)\)[/tex] increases by 1.
- When [tex]\(x = 0\)[/tex], [tex]\(f(x) = -3\)[/tex].
### Analyzing [tex]\(\frac{1}{3} f(x)\)[/tex]:
[tex]\[ \frac{1}{3} f(x) = \frac{1}{3} (x - 3) \][/tex]
Simplifying it, we get:
[tex]\[ \frac{1}{3} f(x) = \frac{1}{3} x - 1 \][/tex]
This transformed function is also a straight line but with a different slope and y-intercept:
- The slope is [tex]\(\frac{1}{3}\)[/tex].
- When [tex]\(x = 0\)[/tex], [tex]\(\frac{1}{3} f(x) = -1\)[/tex].
### Key Points:
- Slope: The new slope is [tex]\(\frac{1}{3}\)[/tex], indicating a less steep incline compared to [tex]\(f(x)\)[/tex].
- Y-intercept: The line intercepts the y-axis at [tex]\(-1\)[/tex].
### Data Points:
Below are some data points calculated for [tex]\(\frac{1}{3} f(x)\)[/tex]:
- When [tex]\(x = -10\)[/tex], [tex]\(f(x) = -13\)[/tex]; therefore [tex]\(\frac{1}{3} f(x) = -4.33...\)[/tex].
- When [tex]\(x = 0\)[/tex], [tex]\(f(x) = -3\)[/tex]; therefore [tex]\(\frac{1}{3} f(x) = -1\)[/tex].
- When [tex]\(x = 10\)[/tex], [tex]\(f(x) = 7\)[/tex]; therefore [tex]\(\frac{1}{3} f(x) = 2.33...\)[/tex].
### Explanation of Points as Plotted on the Graph [tex]\(\frac{1}{3} f(x)\)[/tex]:
- A point [tex]\( (x_1, y_1) \)[/tex] like [tex]\((-10, -4.33)\)[/tex]
- A point [tex]\( (x_2, y_2) \)[/tex] like [tex]\((0, -1)\)[/tex]
- A point [tex]\( (x_3, y_3) \)[/tex] like [tex]\((10, 2.33)\)[/tex]
### Graph Identification:
By comparing the mentioned characteristics and plotting points, we can identify whether the graph shows a line moving slowly upward with the slope of [tex]\(\frac{1}{3}\)[/tex] and intersecting the y-axis at [tex]\(-1\)[/tex] will match our calculated points ideally.
I recommend visually checking the graphs provided (graph 1 or graph 2) to see which one shows:
1. A less steep incline (slope = [tex]\(\frac{1}{3}\)[/tex]).
2. Intercepts the y-axis at -1.
3. Also aligns with data points like (-10, -4.33), (0, -1), and (10, 2.33).
This matches our mathematical derivation of [tex]\(\frac{1}{3}f(x) = \frac{1}{3}x - 1\)[/tex]. Select the graph accordingly that fits the above features.
### Analyzing [tex]\(f(x)\)[/tex]:
Given [tex]\(f(x) = x - 3\)[/tex], this function represents a straight line with a slope of 1 and a y-intercept of [tex]\(-3\)[/tex]. The equation implies that:
- For every increase of 1 in [tex]\(x\)[/tex], [tex]\(f(x)\)[/tex] increases by 1.
- When [tex]\(x = 0\)[/tex], [tex]\(f(x) = -3\)[/tex].
### Analyzing [tex]\(\frac{1}{3} f(x)\)[/tex]:
[tex]\[ \frac{1}{3} f(x) = \frac{1}{3} (x - 3) \][/tex]
Simplifying it, we get:
[tex]\[ \frac{1}{3} f(x) = \frac{1}{3} x - 1 \][/tex]
This transformed function is also a straight line but with a different slope and y-intercept:
- The slope is [tex]\(\frac{1}{3}\)[/tex].
- When [tex]\(x = 0\)[/tex], [tex]\(\frac{1}{3} f(x) = -1\)[/tex].
### Key Points:
- Slope: The new slope is [tex]\(\frac{1}{3}\)[/tex], indicating a less steep incline compared to [tex]\(f(x)\)[/tex].
- Y-intercept: The line intercepts the y-axis at [tex]\(-1\)[/tex].
### Data Points:
Below are some data points calculated for [tex]\(\frac{1}{3} f(x)\)[/tex]:
- When [tex]\(x = -10\)[/tex], [tex]\(f(x) = -13\)[/tex]; therefore [tex]\(\frac{1}{3} f(x) = -4.33...\)[/tex].
- When [tex]\(x = 0\)[/tex], [tex]\(f(x) = -3\)[/tex]; therefore [tex]\(\frac{1}{3} f(x) = -1\)[/tex].
- When [tex]\(x = 10\)[/tex], [tex]\(f(x) = 7\)[/tex]; therefore [tex]\(\frac{1}{3} f(x) = 2.33...\)[/tex].
### Explanation of Points as Plotted on the Graph [tex]\(\frac{1}{3} f(x)\)[/tex]:
- A point [tex]\( (x_1, y_1) \)[/tex] like [tex]\((-10, -4.33)\)[/tex]
- A point [tex]\( (x_2, y_2) \)[/tex] like [tex]\((0, -1)\)[/tex]
- A point [tex]\( (x_3, y_3) \)[/tex] like [tex]\((10, 2.33)\)[/tex]
### Graph Identification:
By comparing the mentioned characteristics and plotting points, we can identify whether the graph shows a line moving slowly upward with the slope of [tex]\(\frac{1}{3}\)[/tex] and intersecting the y-axis at [tex]\(-1\)[/tex] will match our calculated points ideally.
I recommend visually checking the graphs provided (graph 1 or graph 2) to see which one shows:
1. A less steep incline (slope = [tex]\(\frac{1}{3}\)[/tex]).
2. Intercepts the y-axis at -1.
3. Also aligns with data points like (-10, -4.33), (0, -1), and (10, 2.33).
This matches our mathematical derivation of [tex]\(\frac{1}{3}f(x) = \frac{1}{3}x - 1\)[/tex]. Select the graph accordingly that fits the above features.
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