Question
does this graph give a constant rate of change:
x = 1 y = 1
x = 2 y = 4
x = 3 y = 9
x = 4 y = 16
x = 5 y = 25
x = 1 y = 1
x = 2 y = 4
x = 3 y = 9
x = 4 y = 16
x = 5 y = 25
Answers
Answer
Does the graph represent a function?
Is it linear?
Is it a increasing relationship?
Is it proportional?
Is it linear?
Is it a increasing relationship?
Is it proportional?
Answered by
GPT-4o mini
To determine if the graph gives a constant rate of change, we need to look at the differences in y-values as x-values change.
The pairs you provided are as follows:
- For \( x = 1 \), \( y = 1 \)
- For \( x = 2 \), \( y = 4 \)
- For \( x = 3 \), \( y = 9 \)
- For \( x = 4 \), \( y = 16 \)
- For \( x = 5 \), \( y = 25 \)
Let's calculate the differences in \( y \) for each step:
1. From \( x = 1 \) to \( x = 2 \):
- \( \Delta y = 4 - 1 = 3 \)
2. From \( x = 2 \) to \( x = 3 \):
- \( \Delta y = 9 - 4 = 5 \)
3. From \( x = 3 \) to \( x = 4 \):
- \( \Delta y = 16 - 9 = 7 \)
4. From \( x = 4 \) to \( x = 5 \):
- \( \Delta y = 25 - 16 = 9 \)
Now let's summarize the changes:
- The change in \( y \) values is 3, 5, 7, and 9, which are not the same.
Since the differences are not constant, this means the graph does **not** give a constant rate of change. Instead, the change in \( y \) increases as \( x \) increases, indicating that the relationship is quadratic, specifically \( y = x^2 \).
The pairs you provided are as follows:
- For \( x = 1 \), \( y = 1 \)
- For \( x = 2 \), \( y = 4 \)
- For \( x = 3 \), \( y = 9 \)
- For \( x = 4 \), \( y = 16 \)
- For \( x = 5 \), \( y = 25 \)
Let's calculate the differences in \( y \) for each step:
1. From \( x = 1 \) to \( x = 2 \):
- \( \Delta y = 4 - 1 = 3 \)
2. From \( x = 2 \) to \( x = 3 \):
- \( \Delta y = 9 - 4 = 5 \)
3. From \( x = 3 \) to \( x = 4 \):
- \( \Delta y = 16 - 9 = 7 \)
4. From \( x = 4 \) to \( x = 5 \):
- \( \Delta y = 25 - 16 = 9 \)
Now let's summarize the changes:
- The change in \( y \) values is 3, 5, 7, and 9, which are not the same.
Since the differences are not constant, this means the graph does **not** give a constant rate of change. Instead, the change in \( y \) increases as \( x \) increases, indicating that the relationship is quadratic, specifically \( y = x^2 \).
Answered by
GPT-4o mini
Let's analyze each of your questions based on the given points:
### 1. **Does the graph represent a function?**
Yes, the graph represents a function because each input value (x-value) corresponds to exactly one output value (y-value). This is known as the vertical line test in which a vertical line cannot intersect the graph at more than one point. In your case, for \( x = 1, 2, 3, 4, 5 \), there is exactly one \( y \) for each \( x \).
### 2. **Is it linear?**
No, the graph is not linear. A linear function has a constant rate of change (i.e., the slope is constant). As we calculated previously, the rate of change in \( y \) is not constant (it varies: 3, 5, 7, 9), which indicates that the relationship is quadratic (it follows \( y = x^2 \)) rather than linear.
### 3. **Is it an increasing relationship?**
Yes, the relationship is increasing. As the x-values increase from 1 to 5, the corresponding y-values increase from 1 to 25. Therefore, the function is increasing on the interval provided.
### 4. **Is it proportional?**
No, the graph is not proportional. A proportional relationship means that the ratio of \( y \) to \( x \) is constant (i.e., \( y = kx \) for a constant \( k \)). In this case, the ratios of \( y \) values to their corresponding \( x \) values are not constant:
- For \( x = 1, \frac{y}{x} = \frac{1}{1} = 1 \)
- For \( x = 2, \frac{y}{x} = \frac{4}{2} = 2 \)
- For \( x = 3, \frac{y}{x} = \frac{9}{3} = 3 \)
- For \( x = 4, \frac{y}{x} = \frac{16}{4} = 4 \)
- For \( x = 5, \frac{y}{x} = \frac{25}{5} = 5 \)
Since these ratios are not constant but instead increase, the relationship is not proportional.
### Summary:
- **Represents a function**: Yes
- **Linear**: No
- **Increasing relationship**: Yes
- **Proportional**: No
### 1. **Does the graph represent a function?**
Yes, the graph represents a function because each input value (x-value) corresponds to exactly one output value (y-value). This is known as the vertical line test in which a vertical line cannot intersect the graph at more than one point. In your case, for \( x = 1, 2, 3, 4, 5 \), there is exactly one \( y \) for each \( x \).
### 2. **Is it linear?**
No, the graph is not linear. A linear function has a constant rate of change (i.e., the slope is constant). As we calculated previously, the rate of change in \( y \) is not constant (it varies: 3, 5, 7, 9), which indicates that the relationship is quadratic (it follows \( y = x^2 \)) rather than linear.
### 3. **Is it an increasing relationship?**
Yes, the relationship is increasing. As the x-values increase from 1 to 5, the corresponding y-values increase from 1 to 25. Therefore, the function is increasing on the interval provided.
### 4. **Is it proportional?**
No, the graph is not proportional. A proportional relationship means that the ratio of \( y \) to \( x \) is constant (i.e., \( y = kx \) for a constant \( k \)). In this case, the ratios of \( y \) values to their corresponding \( x \) values are not constant:
- For \( x = 1, \frac{y}{x} = \frac{1}{1} = 1 \)
- For \( x = 2, \frac{y}{x} = \frac{4}{2} = 2 \)
- For \( x = 3, \frac{y}{x} = \frac{9}{3} = 3 \)
- For \( x = 4, \frac{y}{x} = \frac{16}{4} = 4 \)
- For \( x = 5, \frac{y}{x} = \frac{25}{5} = 5 \)
Since these ratios are not constant but instead increase, the relationship is not proportional.
### Summary:
- **Represents a function**: Yes
- **Linear**: No
- **Increasing relationship**: Yes
- **Proportional**: No
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