To determine if the relationship shown by the data is linear and to model the data with an equation, let's follow these steps:
1. Given Data Points:
- [tex]\( x = [-9, -5, -1, 3] \)[/tex]
- [tex]\( y = [-2, -7, -12, -17] \)[/tex]
2. Calculate the Slope and Intercept:
We need to find the slope ([tex]\( m \)[/tex]) and the y-intercept ([tex]\( b \)[/tex]) of the best-fit line through the data points. The equation of the line can be written as:
[tex]\[
y = mx + b
\][/tex]
3. Slope and Intercept Calculation:
The slope ([tex]\( m \)[/tex]) and intercept ([tex]\( b \)[/tex]) are determined to be:
[tex]\[
m = -1.25
\][/tex]
[tex]\[
b = -13.25
\][/tex]
4. Form the Equation of the Line:
With the calculated slope and intercept, the equation of the line becomes:
[tex]\[
y = -1.25x + (-13.25)
\][/tex]
Or equivalently:
[tex]\[
y = -1.25x - 13.25
\][/tex]
5. Calculate Predicted [tex]\( y \)[/tex]-Values:
Using the equation [tex]\( y = -1.25x - 13.25 \)[/tex], we can predict the [tex]\( y \)[/tex]-values for the given [tex]\( x \)[/tex]-values:
- For [tex]\( x = -9 \)[/tex]:
[tex]\[
y = -1.25(-9) - 13.25 = 11.25 - 13.25 = -2.00
\][/tex]
- For [tex]\( x = -5 \)[/tex]:
[tex]\[
y = -1.25(-5) - 13.25 = 6.25 - 13.25 = -7.00
\][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[
y = -1.25(-1) - 13.25 = 1.25 - 13.25 = -12.00
\][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[
y = -1.25(3) - 13.25 = -3.75 - 13.25 = -17.00
\][/tex]
So, the predicted [tex]\( y \)[/tex]-values are:
[tex]\[
y_{\text{pred}} = [-2.00, -7.00, -12.00, -17.00]
\][/tex]
6. Conclusion:
Since the predicted [tex]\( y \)[/tex]-values closely match the given [tex]\( y \)[/tex]-values, we can conclude the relationship is linear.
The equation modeling the data is:
[tex]\[
y = -1.25x - 13.25
\][/tex]
This detailed step-by-step solution shows that the relationship between the given data points is indeed linear and it is modeled by the equation [tex]\( y = -1.25x - 13.25 \)[/tex].
