Final answer:
The function f(x) = 2(x-6)² has one x-intercept at (6, 0), due to the repeated root x = 6, and one y-intercept at (0, 72).
Step-by-step explanation:
To find the intercepts for the function f(x) = 2(x-6)², we need to find the points where the graph of the function crosses the x-axis and the y-axis.
X-intercepts (Roots of the function)
To find the x-intercepts, we set f(x) = 0 and solve for x:
0 = 2(x-6)²
The equation suggests that the quadratic term (x-6)² is zero. This happens when x = 6. As this is repeated root, we have just one x-intercept at (6, 0).
Y-intercept
To find the y-intercept, we set x = 0 and solve for f(x):
f(0) = 2(0-6)² = 2(-6)² = 2(36) = 72
Therefore, the y-intercept is at (0, 72).