Question

Sketch the curve. y = x³ − 6x² + 9x.

Answer 1

To sketch the curve of the equation y = x³ − 6x² + 9x, find the y-intercept, x-intercepts, determine the end behavior, and choose additional points on the curve.

Step-by-step explanation:

The equation given, y = x³ − 6x² + 9x, represents the graph of a polynomial function. To sketch the curve, we can use a graphing calculator or follow the steps to identify key points and the behavior of the function.

1. Find the y-intercept by plugging in x = 0: y = 0³ − 6(0)² + 9(0) = 0. Therefore, the y-intercept is (0, 0).
2. Find the x-intercepts by setting y = 0 and solving for x: x³ − 6x² + 9x = 0. Factoring the equation, we get x(x - 3)(x - 3) = 0. This gives us x = 0 and x = 3, so the x-intercepts are (0, 0) and (3, 0).
3. Determine the end behavior by observing the leading term. In this case, the leading term is x³, which means as x approaches positive or negative infinity, y also approaches positive or negative infinity.
4. Choose additional points on the curve by plugging in different x-values. For example, when x = 1, y = 1³ − 6(1)² + 9(1) = 4. So, (1, 4) is another point on the curve.

Using these key points and the behavior, you can sketch the curve of the given equation.