Quadratic and cubic polynomials: graphs overview

Algebra • Graphs

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How can completing the square help to find roots?

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Rewrite the quadratic in completed-square form and then solve a(x - h)^2 + k = 0 by isolating the square and taking square roots.

Key concepts

What you'll likely be quizzed about

Quadratic function definition and basic shape

A quadratic function takes the form y = ax^2 + bx + c. The graph is a parabola. If a > 0 the parabola opens upwards; if a < 0 the parabola opens downwards. Larger |a| produces a steeper parabola; smaller |a| produces a wider parabola.

Roots and intercepts of a quadratic

Roots are x-values where y = 0 and appear as x-intercepts on the graph. The y-intercept occurs at (0, c). Real roots appear when the parabola crosses the x-axis; a repeated root appears when the parabola touches the x-axis at one point. The discriminant b^2 - 4ac determines how many distinct real roots exist: positive for two, zero for one repeated, negative for none.

Turning point and axis of symmetry

The turning point (vertex) is the highest or lowest point of a parabola. The axis of symmetry is the vertical line through the turning point with equation x = -b/(2a). The turning point coordinates arise from rewriting the quadratic in completed-square form y = a(x - h)^2 + k, where (h, k) is the turning point. The value of a determines whether that turning point is a minimum (a > 0) or a maximum (a < 0).

Completing the square to find the turning point

Completing the square converts y = ax^2 + bx + c into y = a(x - h)^2 + k. Divide by a (if a ≠ 0), form (x + b/(2a))^2 by adding and subtracting the required constant, then simplify to find h and k. The result gives the turning point (h, k) directly and simplifies root deduction when further rearranged.

Algebraic deduction of roots

Roots appear from algebraic solutions to ax^2 + bx + c = 0. Factorisation produces roots when factors take the form a(x - r1)(x - r2) = 0. If factorisation is not straightforward, completing the square or the quadratic formula x = (-b ± √(b^2 - 4ac)) / (2a) produces the roots. The algebraic roots match the x-intercepts seen on the graph.

Simple cubic functions: shape and key features

A simple cubic polynomial commonly appears as y = ax^3 + bx^2 + cx + d. The graph shows opposite end-behaviour: if a > 0 the left end falls and the right end rises; if a < 0 the left end rises and the right end falls. A basic cubic like y = x^3 has an inflection point at the origin where curvature changes. Simple cubics can have up to three real roots and up to two turning points; plotting intercepts and a few additional points gives a clear sketch.

Key notes

Important points to keep in mind

Turning point coordinates arise from y = a(x - h)^2 + k as (h, k).

Axis of symmetry for y = ax^2 + bx + c is x = -b/(2a).

Discriminant b^2 - 4ac determines the number of real roots.

Completing the square requires dividing by a when a ≠ 0 before forming a perfect square.

A repeated root means the graph touches but does not cross the x-axis.

For quadratics, calculate one extra point on each side of the vertex to improve sketch accuracy.

Simple cubic end-behaviour depends on the sign of the leading coefficient a.

Factorisation, completing the square and the quadratic formula all give algebraic roots consistent with graph intercepts.

Label roots, turning point and y-intercept when sketching for clarity.

Inflection points on cubics mark where curvature changes, not necessarily where the graph crosses an axis.