ID: c22b5f25 In the \( x y \)-plane, the points \( (-2,3) \) and \( (4,-5) \) lie on the graph of which of the following linear functions? A. \( f(x)=x+5 \) B. \( f(x)=\frac{1}{2} x+4 \) C. \( f(x)=-\frac{4}{3} x+\frac{1}{3} \) D. \( f(x)=-\frac{3}{2} x+1 \)

To determine which linear function goes through the points \((-2,3)\) and \((4,-5)\), we first find the slope using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), which gives us \( m = \frac{-5 - 3}{4 - (-2)} = \frac{-8}{6} = -\frac{4}{3} \). Next, we use the point-slope form \( y - y_1 = m(x - x_1) \) to derive the linear equation. Plugging in one of the points, for instance, \((-2, 3)\), we have: \[ y - 3 = -\frac{4}{3}(x + 2) \] This simplifies to: \[ y = -\frac{4}{3}x + \frac{1}{3} \] Now, if we look at the choices, we find that the answer corresponds to option C, which is \( f(x) = -\frac{4}{3} x + \frac{1}{3} \). --- The concept of slopes and linear functions is not just a theoretical aspect; it plays a significant role in various real-world situations, such as in economics for predicting costs, or in construction when determining the gradient of ramps. Understanding these concepts can help streamline not just calculations but also foster intuitive problem-solving skills in practical applications. Also, one common mistake students make is overlooking the intercept, forgetting to verify if their derived equation’s y-intercept matches the one calculated. Always double-check your calculations by substituting the original points back into the found equation to confirm they satisfy the linear function!