Question

Describe how to graph the solution of
y ≤ −x2 + 2x.

Answer 1

1. Rewrite the parabola in vertex form:
y = −x^2 + 2x = −(x^2 − 2x) = −(x − 1)^2 + 1.
So the vertex is (1, 1) and the axis of symmetry is x = 1.

2. Find intercepts and a few points to plot:
- x-intercepts: solve −x^2 + 2x = 0 → x(−x + 2) = 0 → x = 0 and x = 2 (points (0,0) and (2,0)).
- y-intercept: (0,0).
- Another point, e.g. x = −1 → y = −(−1)^2 + 2(−1) = −1 − 2 = −3 (point (−1, −3)).

3. Draw the parabola y = −x^2 + 2x:
- It opens downward, passes through (0,0), (1,1), (2,0).
- Use a solid curve because the inequality is "≤" (boundary included).

4. Shade the solution region:
- Because y ≤ −x^2 + 2x, shade all points on or below the parabola.
- (Optional check) Test point (1,0): 0 ≤ 1 is true, so the region just below the curve is correct.

Domain: all real x. Range of the solution set: y ≤ 1.