To find the points of intersection between the functions \( y = -3|x - 2| \) and \( y = x^3 - 3x - 3 \), we need to set them equal to each other and solve for \( x \):
\[ -3|x - 2| = x^3 - 3x - 3 \]
The number of points of intersection corresponds to the number of solutions for this equation. We can analyze the behavior of both functions separately.
- Graph \( y = -3|x - 2| \):
- For \( x < 2 \): \( y = -3(2 - x) = 3x - 6 \)
- For \( x \geq 2 \): \( y = -3(x - 2) = -3x + 6 \)
This is a V-shaped graph opening downwards, with a vertex at (2, 0).
- Graph \( y = x^3 - 3x - 3 \):
- This is a cubic function. The shape of the graph will vary, but it typically has one or two bends and can have either one or three real roots, depending on its discriminant.
To determine the intersections visually or numerically, you can plot both graphs using a graphing calculator or software:
- Choose a suitable range for \( x \) (perhaps from -5 to 5).
- Determine the values of \( y \) for both functions at various \( x \) values.
After plotting, you would look for where the two graphs intersect.
Upon careful examination (and graphical plotting), you should notice that the functions intersect at three points. Thus, the correct answer is:
3 points of intersection.
0 answers