Determine each feature of the graph of the given function. f(x)=(-(2x+6)^2(x-3))/(3(2x-4)(2x+3)(x-3)) Answer Attempt2 out of 2 # of Horizontal Asymptotes: square # of Holes: square # of Vertical Asymptotes: square # of x-intercepts: square # of y-intercepts: square

To determine the features of the graph of the given function, we need to analyze the function's behavior in terms of horizontal asymptotes, holes, vertical asymptotes, x-intercepts, and y-intercepts.<br /><br />1. Horizontal Asymptotes:<br /> - A horizontal asymptote is a horizontal line that the graph approaches as x approaches positive or negative infinity.<br /> - To find the horizontal asymptotes, we need to look at the behavior of the function as x approaches positive or negative infinity.<br /> - In this case, the function is a rational function, and the degree of the numerator is greater than the degree of the denominator.<br /> - Therefore, there are no horizontal asymptotes.<br /><br />2. Holes:<br /> - A hole is a point on the graph where the function is undefined due to a factor in the numerator and denominator that can be canceled out.<br /> - In this case, the function has a factor of (x-3) in both the numerator and denominator, which can be canceled out.<br /> - Therefore, there is a hole at x = 3.<br /><br />3. Vertical Asymptotes:<br /> - A vertical asymptote is a vertical line where the function approaches positive or negative infinity as x approaches a specific value.<br /> - In this case, the function has factors of (2x-4) and (2x+3) in the denominator, which can cause vertical asymptotes.<br /> - Therefore, there are vertical asymptotes at x = 2 and x = -3/2.<br /><br />4. X-intercepts:<br /> - An x-intercept is a point where the graph intersects the x-axis, i.e., where the function is equal to zero.<br /> - In this case, the function is equal to zero when the numerator is equal to zero.<br /> - The numerator is (-(2x+6)^2)(x-3), which is equal to zero when x = -3 or x = 3.<br /> - Therefore, there are two x-intercepts at x = -3 and x = 3.<br /><br />5. Y-intercepts:<br /> - A y-intercept is a point where the graph intersects the y-axis, i.e., where the function is equal to zero when x = 0.<br /> - In this case, the function is equal to zero when x = 0.<br /> - Substituting x = 0 into the function, we get:<br /> f(0) = -(2(0)+6)^2(0-3)/(3(2(0)-4)(2(0)+3)(0-3))<br /> f(0) = -(-6)^2(-3)/(3(-4)(3)(-3))<br /> f(0) = -36/(-36) = 1<br /> - Therefore, there is a y-intercept at y = 1.<br /><br />In summary:<br />- Number of Horizontal Asymptotes: 0<br />- Number of Holes: 1 (at x = 3)<br />- Number of Vertical Asymptotes: 2 (at x = 2 and x = -3/2)<br />- Number of X-intercepts: 2 (at x = -3 and x = 3)<br />- Number of Y-intercepts: 1 (at y = 1)