62.6 Quadratic Graph Interpretation

Quadratic Graph Interpretation is an alternative approach to solving a quadratic inequality by reading the solution directly from the shape and position of the corresponding parabola, using the curve's position above or below the horizontal axis in place of the algebraic sign testing performed in interval analysis.

Quadratic Graph above the Axis

Regions Where the Curve Lies Above the Axis

Any region of the graph where the parabola lies above the horizontal axis corresponds to input values for which the quadratic expression is positive.

Using This to Solve a Positive Inequality

For an inequality requiring the quadratic expression to be greater than zero, the input values corresponding to this above-the-axis region are read directly as the solution, without needing separate sign testing.

Quadratic Graph below the Axis

Regions Where the Curve Lies Below the Axis

Any region of the graph where the parabola lies below the horizontal axis corresponds to input values for which the quadratic expression is negative.

Using This to Solve a Negative Inequality

For an inequality requiring the quadratic expression to be less than zero, the input values corresponding to this below-the-axis region are read directly as the solution, without needing separate sign testing.

Horizontal Intercepts as Boundaries

The Graphical Role of Intercepts

The horizontal intercepts of the parabola serve the same role graphically that boundary values serve algebraically, marking the exact points where the curve transitions between lying above and below the axis.

Consistency between the Two Approaches

Because these intercepts are found by solving the same boundary equation used in interval sign analysis, both approaches rely on identical underlying values, differing only in whether that information is read from a picture or tested algebraically.

Upward-Parabola Exterior Positivity

Sign Behavior for an Upward-Opening Curve

For a parabola that opens upward and crosses the horizontal axis at two points, the exterior regions, to the left of the smaller intercept and to the right of the larger intercept, lie above the axis and are positive.

Why This Pattern Holds

Because an upward-opening curve rises away from its vertex in both directions, and its vertex lies between the two intercepts below the axis, the curve necessarily rises back above the axis on both outer sides.

Upward-Parabola Interior Negativity

Sign Behavior between the Intercepts

For the same upward-opening parabola, the interior region, between the two intercepts, lies below the axis and is negative.

Why This Pattern Holds

The vertex of an upward-opening parabola between two intercepts must lie below the axis, since the curve must dip down and back up to cross the axis at two separate points, placing the entire interior region beneath that axis.

Downward-Parabola Sign Reversal

Sign Behavior for a Downward-Opening Curve

For a parabola that opens downward and crosses the horizontal axis at two points, the sign pattern is entirely reversed compared to the upward-opening case: the exterior regions are negative, and the interior region is positive.

Why the Pattern Reverses

Because a downward-opening curve falls away from its vertex in both directions, with the vertex sitting above the axis between the two intercepts, the entire relationship between above and below the axis is flipped compared to the upward-opening case.

Quadratic Graph Solution Check

Confirming Agreement with Algebraic Analysis

The solution read from the graph is checked against the solution produced by interval sign analysis, confirming that both approaches identify the same regions and the same boundary treatment.

Why This Cross-Check Is Valuable

Because the graphical and algebraic approaches rely on entirely different reasoning — visual position versus computed sign — agreement between them provides strong confirmation that the final solution to the inequality is correct.