5.Where would the graph of y=x^2 and y=x^3 intersect?
At x = -1
At all x>1
At x=0 and x=1
At all points between x=0 and x=1
The graphs of \(y = x^2\) and \(y = x^3\) intersect at x=0 and x=1.[web:1][web:6]
Solving y = x² = x³
Set the equations equal: \(x^3 = x^2\). Rearrange to \(x^3 – x^2 = 0\), or \(x^2(x – 1) = 0\). Solutions are x=0 (double root) and x=1. At these points, y=0 and y=1 respectively, confirming intersections at (0,0) and (1,1).[web:1]
\(x^3 – x^2 = 0\)
\(x^2(x – 1) = 0\)
\(x = 0\) or \(x = 1\)
\(x^2(x – 1) = 0\)
\(x = 0\) or \(x = 1\)
Option Analysis
- At x = -1: y = (-1)² = 1, but y = (-1)³ = -1. Values differ, so no intersection.[web:1]
- At all x>1: For x=2, y = 4 vs y = 8; curves diverge as cubic grows faster. No continuous intersection.[web:6]
- At x=0 and x=1: Verified algebraically; both functions equal at these discrete points.[web:1]
- At all points between x=0 and x=1: At x=0.5, y = 0.25 vs y = 0.125; curves cross only at endpoints, not continuously.
Graph Behavior Insights
The parabola y = x² opens upward symmetrically. The cubic y = x³ passes through origin with inflection. They touch at origin, separate between 0 and 1 (x² above), then x³ overtakes for x>1.[web:6]
Correct Answer: At x=0 and x=1
