Previous Problem Problem List Next Problem (1 point) Fit a linear function of the form $f(t)=c_{0}+c_{1} t$ to the data points $(-4,23),(0,-3),(4,-17)$, using least squares. \[ f(t)= \]

Step 1 ：We are given the data points (-4,23), (0,-3), and (4,-17). We are asked to fit a linear function of the form \(f(t)=c_{0}+c_{1} t\) to these data points using the least squares method.

Step 2 ：The least squares method involves minimizing the sum of the squared residuals. The residuals are the differences between the observed and predicted values of the dependent variable.

Step 3 ：We define the function for the sum of squared residuals as \( \sum_{i=1}^{n} (y_{i} - (c_{0} + c_{1}x_{i}))^{2} \), where \(x_{i}\) and \(y_{i}\) are the coordinates of the data points, and \(c_{0}\) and \(c_{1}\) are the parameters of the linear function we are trying to fit.

Step 4 ：We use a numerical optimization method to find the values of \(c_{0}\) and \(c_{1}\) that minimize the sum of squared residuals. We provide an initial guess of [0,0] for \(c_{0}\) and \(c_{1}\).

Step 5 ：The optimization method returns the values of \(c_{0}\) and \(c_{1}\) that minimize the sum of squared residuals. The values are \(c_{0} = 1\) and \(c_{1} = -5\).

Step 6 ：Therefore, the linear function that fits the data points using the least squares method is \(f(t) = 1 - 5t\).

Step 7 ：Final Answer: The linear function that fits the data points using the least squares method is \(f(t) = \boxed{1 - 5t}\).