Quadratic Equations and Graphing
Parabolas, roots, and the discriminant
A quadratic in standard form is y = ax² + bx + c, and its graph is a parabola that opens upward when a is positive and downward when a is negative. The roots, zeros, or x-intercepts are the solutions of ax² + bx + c = 0 and can be found by factoring, completing the square, or the quadratic formula x = (−b ± √(b² − 4ac)) / (2a).
The discriminant b² − 4ac predicts the number of real roots: positive gives two, zero gives exactly one, negative gives none, meaning the parabola never crosses the x-axis. The vertex sits at x = −b/(2a), and that vertical line is the axis of symmetry. Vertex form y = a(x − h)² + k displays the vertex as (h, k).
When graphing, plot the intercepts, the vertex, and one or two symmetric points for shape. Lines are graphed from slope-intercept form by plotting b on the y-axis and stepping with the slope.
Traps: forgetting the ± sign in the quadratic formula, losing the negative in −b/(2a), applying the zero product property only when one side equals zero, and confusing "no real solutions" with "no graph."
Worked example: x² − 5x + 6 = 0 factors as (x − 2)(x − 3) = 0, so x = 2 or x = 3. The discriminant is 25 − 24 = 1, positive, confirming two real roots. The vertex lies at x = 5/2 = 2.5.