Question
is the table Linear, Quadratic, Exponential?
x y
1 5
2 9
3 13
4 17
5 21
x y
1 5
2 9
3 13
4 17
5 21
Answers
Answered by
GPT-4o mini
To determine whether the table represents a linear, quadratic, or exponential relationship, we can analyze the pattern of the values provided.
Given the data:
```
x | y
-----
1 | 5
2 | 9
3 | 13
4 | 17
5 | 21
```
First, we will check if the relationship is linear by calculating the differences between consecutive \(y\) values:
- \(9 - 5 = 4\)
- \(13 - 9 = 4\)
- \(17 - 13 = 4\)
- \(21 - 17 = 4\)
The first differences are constant (all equal to 4), which indicates that the relationship is linear.
Next, we can express the linear relationship. The general form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In this case, the slope \(m\) is 4.
To find \(b\), we can substitute one of the points into the equation. Using the point (1, 5):
\[
5 = 4(1) + b \implies b = 5 - 4 = 1
\]
Thus, the linear equation that represents the relationship is:
\[
y = 4x + 1
\]
Since the differences between \(y\) values are constant, and we derived a linear equation, we can conclude that the relationship is linear.
The table is **linear**. It is not quadratic (which would have a second difference that is constant) or exponential (which would show a multiplicative pattern).
Given the data:
```
x | y
-----
1 | 5
2 | 9
3 | 13
4 | 17
5 | 21
```
First, we will check if the relationship is linear by calculating the differences between consecutive \(y\) values:
- \(9 - 5 = 4\)
- \(13 - 9 = 4\)
- \(17 - 13 = 4\)
- \(21 - 17 = 4\)
The first differences are constant (all equal to 4), which indicates that the relationship is linear.
Next, we can express the linear relationship. The general form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In this case, the slope \(m\) is 4.
To find \(b\), we can substitute one of the points into the equation. Using the point (1, 5):
\[
5 = 4(1) + b \implies b = 5 - 4 = 1
\]
Thus, the linear equation that represents the relationship is:
\[
y = 4x + 1
\]
Since the differences between \(y\) values are constant, and we derived a linear equation, we can conclude that the relationship is linear.
The table is **linear**. It is not quadratic (which would have a second difference that is constant) or exponential (which would show a multiplicative pattern).