2. Indicate all points where graph of y=x2 and y=x3 intersect?
a. At x=y=0
b. At x=y=1
c. At x=y=0 and x=y=1
d. At x=y=-1

Intersection of y = x² and y = x³

The graphs of y = x² and y = x³ intersect at two points: (0, 0) and (1, 1).

Correct Option: c. At x = y = 0 and x = y = 1.

Step-by-Step Solution

To find the intersection points, set the two expressions for y equal:

x² = x³

Rearrange to one side: x³ – x² = 0

Factorize: x²(x – 1) = 0

Now solve each factor:

• x² = 0 ⇒ x = 0
• x – 1 = 0 ⇒ x = 1

For each value of x, substitute into either equation (both give the same y at intersection):

• For x = 0: y = 0² = 0 → Point (0, 0)
• For x = 1: y = 1² = 1 → Point (1, 1)

Thus, the graphs intersect at (0, 0) and (1, 1), which means x = y = 0 and x = y = 1.

Explanation of Each Option

Option a: At x = y = 0
Includes point (0, 0), which is an intersection, but misses (1, 1), so it is incomplete and incorrect.

Option b: At x = y = 1
Includes point (1, 1), also an intersection, but omits (0, 0); hence incomplete and incorrect.

Option c: At x = y = 0 and x = y = 1
Lists both intersection points (0, 0) and (1, 1); therefore, this is the correct answer.

Option d: At x = y = –1
For x = –1: y = x² = 1 and y = x³ = –1 — both yield different y values, so no intersection at (–1, –1). Thus, this option is incorrect.