Describe how to graph the solution of $y \leq -x^2 + 2x$.

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05-11-2026
Mathematics

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Describe how to graph the solution of $y \leq -x^2 + 2x$.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-05T10:14:55-05:00

Identify the boundary of the region as the parabola $y = -x^2 + 2x$.
Find key points: $x$-intercepts at $(0, 0)$ and $(2, 0)$, and the vertex at $(1, 1)$.
Draw the boundary parabola as a solid line because the inequality is $\leq$.
Test a point like $(1, 0)$ to confirm that the region to be shaded is below and inside the parabola.

Explanation

Analyzing the Inequality and Its Boundary

To graph the solution set of the inequality $y \leq -x^2 + 2x$, we first examine the boundary, which is given by the equation $y = -x^2 + 2x$. This equation represents a parabola in the Cartesian plane. Since the coefficient of the $x^2$ term is negative, the parabola opens downwards. The inequality symbol $\leq$ indicates that we must include the points on the parabola itself and all points in the region below it.

Finding Key Points of the Parabola

We identify key features of the parabola to facilitate graphing. The $x$-intercepts occur when $y = 0$:
\[0 = -x^2 + 2x = -x(x - 2)\]
This gives us intercepts at $x = 0$ and $x = 2$. The vertex of a parabola $y = ax^2 + bx + c$ is found at $x_v = -\frac{b}{2a}$. Here, $a = -1$ and $b = 2$, so $x_v = -\frac{2}{2(-1)} = 1$. Substituting $x = 1$ into the equation gives $y_v = -(1)^2 + 2(1) = 1$. The vertex is at $(1, 1)$.

Determining the Region and Finalizing the Graph

Since the inequality uses $\leq$, the boundary parabola $y = -x^2 + 2x$ is drawn as a solid line. Next, we determine which region to shade. We select a test point not on the parabola, such as $(1, 0)$, which is below the vertex. Plugging this into the inequality:
\[0 \leq -(1)^2 + 2(1) = 1\]
Since $0 \leq 1$ is true, the point $(1, 0)$ satisfies the inequality. Therefore, we shade the region inside and below the downward-opening parabola, including the boundary line.

Examples

Understanding how to graph quadratic inequalities is useful in fields like engineering and economics, where you might need to determine the feasible region for a project. For instance, if a manufacturing process is limited by a constraint described by a parabolic boundary—such as an efficiency limit based on speed—the shaded region represents all the operational combinations of parameters that are within the allowed safe or efficient threshold. This visual representation helps decision-makers identify valid zones for resource allocation.