Write A Quadratic Inequality Represented By The Graph

A quadratic inequality is an expression that involves a quadratic expression and an inequality sign, such as <, >, ≤, or ≥. The graph of a quadratic inequality represents all the points (x, y) that satisfy the inequality. Understanding how to write a quadratic inequality from its graph is an essential skill in algebra, as it allows you to translate visual information into a mathematical statement.

To write a quadratic inequality from a graph, you must first identify the type of parabola shown. The parabola can open upward or downward, and the graph may be shaded above or below the curve. The shading indicates the region where the inequality is true. If the parabola is dashed, the points on the parabola do not satisfy the inequality; if it is solid, the points on the parabola are included.

The general form of a quadratic inequality is ax² + bx + c < 0, ax² + bx + c > 0, ax² + bx + c ≤ 0, or ax² + bx + c ≥ 0. The direction of the parabola is determined by the sign of the coefficient a: if a is positive, the parabola opens upward; if a is negative, it opens downward. The position of the parabola relative to the x-axis and the direction of shading will tell you which inequality sign to use.

To write the inequality, follow these steps:

1. Identify the equation of the parabola: Find the standard form y = ax² + bx + c or vertex form y = a(x - h)² + k that matches the graph.
2. Determine the direction of the inequality: If the region above the parabola is shaded, use > or ≥; if the region below is shaded, use < or ≤.
3. Decide on the inclusion of the parabola: If the parabola is solid, use ≤ or ≥; if it is dashed, use < or >.
4. Write the inequality: Replace y with the expression from the equation and add the appropriate inequality sign.

For example, consider a parabola that opens upward with vertex at (0, -4), x-intercepts at x = -2 and x = 2, and the region below the parabola shaded. The equation of the parabola is y = x² - 4. Since the region below the parabola is shaded and the parabola is solid, the inequality is y ≤ x² - 4.

Another example is a parabola that opens downward with vertex at (0, 3), x-intercepts at x = -1 and x = 1, and the region above the parabola shaded. The equation is y = -x² + 3. Since the region above the parabola is shaded and the parabola is solid, the inequality is y ≥ -x² + 3.

Sometimes, you may need to solve the inequality for x. To do this, set the quadratic expression equal to zero and find the roots. These roots divide the number line into intervals. Test a point from each interval to see if it satisfies the inequality. The solution set is the union of all intervals where the inequality is true.

The graph of a quadratic inequality can be analyzed using the discriminant, D = b² - 4ac. If D > 0, the parabola intersects the x-axis at two points; if D = 0, it touches the x-axis at one point; if D < 0, it does not intersect the x-axis. The sign of D helps determine the number of solutions to the inequality.

In summary, writing a quadratic inequality from its graph involves identifying the equation of the parabola, determining the direction and inclusion of the inequality, and expressing it in the appropriate form. By following these steps and practicing with various examples, you can become proficient in translating graphs into quadratic inequalities and vice versa.

To further solidify understanding, let’s explore solving quadratic inequalities algebraically and their practical implications. Once the roots of the quadratic equation $ ax^2 + bx + c = 0 $ are found, they partition the number line into intervals. Testing a sample point from each interval in the original inequality reveals where the expression $ ax^2 + bx + c $ is positive or negative. For instance, if the inequality is $ x^2 - 5x + 6 > 0 $, factoring gives roots at $ x = 2 $ and $ x = 3 $. Testing intervals: for $ x < 2 $, choose $ x = 1 $ (yields $ 2 > 0 $, true); for $ 2 < x < 3 $, choose $ x = 2.5 $ (yields $ -0.25 > 0

When the test pointconfirms that an interval satisfies the inequality, we keep that interval in the solution set; otherwise we discard it. The process is identical whether the inequality is strict ( >  or < ) or non‑strict ( ≥  or ≤ ); the only difference lies in whether the boundary points—where the quadratic equals zero—are retained.

Example 1 – Strict inequality

Solve (x^{2}-5x+6>0).

1. Factor: ((x-2)(x-3)>0).

2. Roots: (x=2) and (x=3).

3. Intervals: ((-\infty,2),;(2,3),;(3,\infty)).

4. Choose test values:

   • (x=1) → ((1-2)(1-3)=(-1)(-2)=2>0) → satisfies.
   • (x=2.5) → ((2.5-2)(2.5-3)=(0.5)(-0.5)=-0.25<0) → does not satisfy.
   • (x=4) → ((4-2)(4-3)=2\cdot1=2>0) → satisfies.

Hence the solution is ((-\infty,2)\cup(3,\infty)). Because the inequality is strict, the endpoints 2 and 3 are excluded.

Example 2 – Non‑strict inequality

Solve ( -x^{2}+4x-3\ge0).

1. Rewrite as ( -(x^{2}-4x+3)\ge0) or (x^{2}-4x+3\le0).

2. Factor: ((x-1)(x-3)\le0). 3. Roots: (x=1) and (x=3).

3. Intervals: ((-\infty,1),;(1,3),;(3,\infty)).

4. Test points:

   • (x=0) → ((0-1)(0-3)=(-1)(-3)=3>0) → does not satisfy the “≤0” condition.
   • (x=2) → ((2-1)(2-3)=1\cdot(-1)=-1\le0) → satisfies.
   • (x=4) → ((4-1)(4-3)=3\cdot1=3>0) → does not satisfy.

Because the inequality is non‑strict, the points where the expression equals zero (1 and 3) are included. The solution set is ([1,3]).

Using a sign chart

A sign chart offers a visual shortcut. After locating the zeros, draw a number line, mark the zeros, and write the sign of each factor in the intervals. Multiply the signs across each interval to obtain the overall sign of the quadratic. This method scales elegantly to higher‑degree polynomials and to expressions that are products of several linear factors.

When the quadratic does not factor nicely

If factoring is impractical, the quadratic formula provides the roots:

[ x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}. ]

Even when the roots are irrational or complex, the same interval‑testing strategy applies. If the discriminant (D=b^{2}-4ac) is negative, the quadratic never touches the x‑axis; its sign is constant across the entire real line. In that case, the inequality is either always true (if the constant sign matches the desired inequality) or always false.

Practical implications

Quadratic inequalities appear frequently in real‑world modeling. For instance:

• Optimization constraints – In operations research, a profit function might be modeled as a downward‑opening parabola; requiring profit to be non‑negative translates to a quadratic inequality that defines the feasible production levels.
• Physics problems – The trajectory of a projectile is described by a quadratic equation; determining the time intervals during which the height exceeds a certain threshold involves solving a quadratic inequality.
• Engineering tolerances – When a material’s stress–strain relationship follows a quadratic trend, safety standards may stipulate that stress must stay below a certain curve, leading to an inequality of the form ( \sigma(x)\le \sigma_{\text{allowable}} ).

In each case, translating a graphical representation into an algebraic inequality enables precise quantitative analysis and decision‑making.

Conclusion

Writing a quadratic inequality from its graph is a systematic skill that blends visual interpretation with algebraic manipulation. By extracting the underlying equation, discerning whether the boundary is included, and then analyzing the sign of the expression across the intervals determined by its roots, one can accurately capture the region defined by the inequality. Mastery of this process not only reinforces conceptual understanding of parabolas and their properties but also equips students with a versatile tool for solving practical problems in science, engineering, and economics. Continual practice—starting with simple factored forms and progressing to more complex, unfactored quadratics—builds confidence and fluency, allowing the graph‑to‑inequality translation to become second nature.