To determine if the given data represents a linear function, we need to check if the data points can be fitted to a linear equation of the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Let's look at the data provided:
[tex]\[ x = [1, 3, 5, 7] \][/tex]
[tex]\[ y = [-2, 1, 4, 7] \][/tex]
### Step-by-Step Solution
1. Calculate the Differences in [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
- For [tex]\( x_i \)[/tex] values: [tex]\( 3 - 1 = 2 \)[/tex], [tex]\( 5 - 3 = 2 \)[/tex], [tex]\( 7 - 5 = 2 \)[/tex]
- For corresponding [tex]\( y_i \)[/tex] values: [tex]\( 1 - (-2) = 3 \)[/tex], [tex]\( 4 - 1 = 3 \)[/tex], [tex]\( 7 - 4 = 3 \)[/tex]
2. Check if the Ratios of Differences are Constant:
- The differences in [tex]\( x \)[/tex] values are constant: 2.
- The differences in [tex]\( y \)[/tex] values are also constant: 3.
3. Calculate the Slope ([tex]\(m\)[/tex]):
- [tex]\( m = \frac{\Delta y}{\Delta x} = \frac{3}{2} \)[/tex]
4. Form the Linear Equation:
- Use one of the points to solve for the y-intercept ([tex]\(b\)[/tex]):
- For [tex]\( (x, y) = (1, -2) \)[/tex]:
[tex]\( y = mx + b \)[/tex]
[tex]\[ -2 = \frac{3}{2}(1) + b \][/tex]
[tex]\[ -2 = \frac{3}{2} + b \][/tex]
[tex]\[ b = -2 - \frac{3}{2} = -\frac{4}{2} - \frac{3}{2} = -\frac{7}{2} \][/tex]
- Thus, the linear equation is:
[tex]\[ y = \frac{3}{2}x - \frac{7}{2} \][/tex]
5. Verify the Linear Equation with Other Points:
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ y = \frac{3}{2}(3) - \frac{7}{2} = \frac{9}{2} - \frac{7}{2} = 1 \][/tex]
- For [tex]\( x = 5 \)[/tex]:
[tex]\[ y = \frac{3}{2}(5) - \frac{7}{2} = \frac{15}{2} - \frac{7}{2} = 4 \][/tex]
- For [tex]\( x = 7 \)[/tex]:
[tex]\[ y = \frac{3}{2}(7) - \frac{7}{2} = \frac{21}{2} - \frac{7}{2} = 7 \][/tex]
Since all the calculated [tex]\( y \)[/tex]-values match the given [tex]\( y \)[/tex]-values, the data points can indeed be fitted to a linear equation.
Therefore, the answer is:
Yes
