Find the \( x \)-intercepts and \( y \)-intercept of the following function. \[ f(x)=(x-4)(x+1)(x+3) \] Select the correct answer below: \( x \)-intercepts: \( (3,0),(1,0) \), and \( (-4,0) \cdot y \)-intercept: \( (0,0) \) \( x \)-intercepts: \( (3,0),(1,0) \), and \( (-4,0) \cdot y \)-intercept: \( (0,12) \) \( x \)-intercepts: \( (-3,0),(-1,0) \), and \( (4,0) \cdot y \)-intercept: \( (0,12) \) \( x \)-intercepts: \( (-3,0),(-1,0) \), and \( (4,0) \cdot y \)-intercept: \( (0,0) \) \( x \)-intercepts: \( (-3,0),(-1,0) \), and \( (4,0) \cdot y \)-intercept: \( (0,-12) \) \( x \)-intercepts: \( (3,0),(1,0) \), and \( (-4,0) \cdot y \)-intercept: \( (0,-12) \)

To find the \( x \)-intercepts of the function \( f(x) = (x-4)(x+1)(x+3) \), we set \( f(x) = 0 \). This means we solve for when each factor equals zero: 1. \( x - 4 = 0 \) gives \( x = 4 \), 2. \( x + 1 = 0 \) gives \( x = -1 \), 3. \( x + 3 = 0 \) gives \( x = -3 \). So, the \( x \)-intercepts are \( (-3,0), (-1,0), (4,0) \). Next, to find the \( y \)-intercept, we evaluate \( f(0) \): \[ f(0) = (0-4)(0+1)(0+3) = (-4)(1)(3) = -12. \] Thus, the \( y \)-intercept is \( (0,-12) \). The correct answer is: \( x \)-intercepts: \( (-3,0),(-1,0), (4,0) \) and \( y \)-intercept: \( (0,-12) \). --- You just dived into intercepts! Did you know that the number of intercepts can signal the function's behavior? For polynomial functions like this one, the roots (or \( x \)-intercepts) reveal fundamental properties about the graph—such as where it crosses the x-axis. It's like a treasure map for graphing! Additionally, to plot this function more accurately, consider using a graphing calculator or an online graphing tool. It can help visualize where the function crosses the axes and what the overall shape looks like. This way, you can ensure you don’t miss other fascinating behaviors of the function between those intercepts!