Final answer:
To find the x-intercepts of the function f(x) = x³ - 9x² + 20x - 12, we can factor the function by grouping and solve for x. The x-intercepts are x = 9. To find the y-intercept, we substitute x = 0 into the function and solve for y. The y-intercept is y = -12.
Step-by-step explanation:
To find the x-intercepts of the function f(x) = x³ - 9x² + 20x - 12, we need to set the function equal to zero and solve for x. This can be done by factoring the function or using the quadratic formula. In this case, we can factor the function by grouping.
First, we group the first two terms and the last two terms:
(x³ - 9x²) + (20x - 12)
Next, we factor out the greatest common factor from each group:
x²(x - 9) + 4(5x - 3)
Now, we can factor out (x - 9) from the first group and 4 from the second group:
(x - 9)(x² + 4) + 4(5x - 3)
Setting this equal to zero, we get:
(x - 9)(x² + 4) + 20x - 12 = 0
From here, we can solve for x using the zero-product property that states if a*b = 0, then either a = 0 or b = 0.
Setting each factor equal to zero, we have:
x - 9 = 0 or x² + 4 = 0
Solving each equation, we find:
x = 9 or x² = -4 (which has no real solutions).
Therefore, the x-intercepts of the graph of f(x) are x = 9.
To find the y-intercept, we substitute x = 0 into the function and solve for y:
f(0) = (0)³ - 9(0)² + 20(0) - 12
f(0) = -12
Therefore, the y-intercept of the graph of f(x) is y = -12.