Consider the graph of the functions \( f(x)=(x-5)^{2} \) and \( g(x)=-3 x+13 \). What are the points at which \( f(x)=g(x) \) ?

To find the points where \( f(x) = g(x) \), we set the two functions equal to each other and solve for \( x \): \[ (x - 5)^2 = -3x + 13 \] **Step 1: Expand and Simplify** \[ x^2 - 10x + 25 = -3x + 13 \] Bring all terms to one side of the equation: \[ x^2 - 10x + 25 + 3x - 13 = 0 \] \[ x^2 - 7x + 12 = 0 \] **Step 2: Factor the Quadratic** \[ (x - 3)(x - 4) = 0 \] So, the solutions for \( x \) are: \[ x = 3 \quad \text{and} \quad x = 4 \] **Step 3: Find the Corresponding \( y \)-Values** Using \( g(x) = -3x + 13 \): - For \( x = 3 \): \[ y = -3(3) + 13 = -9 + 13 = 4 \] - For \( x = 4 \): \[ y = -3(4) + 13 = -12 + 13 = 1 \] **Final Answer:** The functions \( f(x) \) and \( g(x) \) intersect at the points \( (3, 4) \) and \( (4, 1) \). **Graphical Representation:**  *(Note: The image link is illustrative. Please plot the functions for a visual representation.)* **Conclusion:** The points of intersection are \( (3, 4) \) and \( (4, 1) \). **Answer:** After simplifying, the graphs meet at (3 , 4) and at (4, 1). Thus, the points are (3, 4) and (4, 1).