The graph of y = x$^2$ and y = x intersect at which of these points?

Analyze the question: We need the points where the curve y = x² meets the line y = x. At an intersection point, both equations give the same y-value, so we set them equal.

Concept: To find where two graphs intersect, equate the two expressions for y, solve for x, then substitute back to find y.

Solution:

1. Set the equations equal: x² = x

2. Rearrange: x² − x = 0

3. Factorise: x(x − 1) = 0

4. Solve: x = 0 or x = 1

5. Find y using y = x: • When x = 0, y = 0, giving (0, 0) • When x = 1, y = 1, giving (1, 1)

Check: For (1, 1): y = x² gives 1² = 1 ✓ and y = x gives 1 ✓. For (0, 0): both give 0 ✓.

Checking the other options:

• B. (0, 1) is not on y = x, so it is wrong.
• C. (1, 0) is not on y = x, so it is wrong.
• D. This lists only (0, 0) twice, so it misses the second intersection point.

Answer: A. (0,0), (1,1)

Tip: Whenever you're asked for the intersection of two graphs, equate the two equations and solve. Never divide both sides by x in x² = x, because you would lose the solution x = 0. Always factorise instead.