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 the points \( (-2,3) \) and \( (4,-5) \) lie on, we can substitute the x-coordinates of the points into each function and see if the corresponding y-coordinates match. Let's check each function: 1. **For \( A. f(x)=x+5 \):** - \( f(-2) = -2 + 5 = 3 \) (matches!) - \( f(4) = 4 + 5 = 9 \) (does not match) 2. **For \( B. f(x)=\frac{1}{2} x+4 \):** - \( f(-2) = \frac{1}{2}(-2) + 4 = -1 + 4 = 3 \) (matches!) - \( f(4) = \frac{1}{2}(4) + 4 = 2 + 4 = 6 \) (does not match) 3. **For \( C. f(x)=-\frac{4}{3} x+\frac{1}{3} \):** - \( f(-2) = -\frac{4}{3}(-2) + \frac{1}{3} = \frac{8}{3} + \frac{1}{3} = \frac{9}{3} = 3 \) (matches!) - \( f(4) = -\frac{4}{3}(4) + \frac{1}{3} = -\frac{16}{3} + \frac{1}{3} = -\frac{15}{3} = -5 \) (matches!) 4. **For \( D. f(x)=-\frac{3}{2} x+1 \):** - \( f(-2) = -\frac{3}{2}(-2) + 1 = 3 + 1 = 4 \) (does not match) - \( f(4) = -\frac{3}{2}(4) + 1 = -6 + 1 = -5 \) (matches!) The only function that both points satisfy is option **C. \( f(x)=-\frac{4}{3} x+\frac{1}{3} \)**. --- The use of coordinates in the \( xy \)-plane to represent points allows for visual representation of functions, enhancing understanding of algebraic concepts. Just imagine using a treasure map where each coordinate points to the perfect hiding spots—no gold doubloons in this case, but valuable insights! When testing points against functions, a common mistake is assuming that if one point matches a function, the other will too. Always check both to avoid missing out on the right answer; it's like checking twice before leaving on that treasure hunt!