β Unit 1: Functions, relations and graphs
How are quadratic functions analysed?
Sketch and analyse quadratic functions in standard, factored and turning-point form, including finding vertex, axis of symmetry, intercepts and using the discriminant to classify roots
A focused answer to the VCE Maths Methods Unit 1 dot point on quadratic functions. Sketches $y = ax^2 + bx + c$, converts between forms, finds the vertex from $x = -b/(2a)$, applies the discriminant $b^2 - 4ac$, and works the VCAA SAC-style turning-point and roots problem.
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What this dot point is asking
VCAA wants you to sketch and analyse quadratic functions in their three standard forms, find vertex and intercepts, and use the discriminant to classify roots.
The three forms
Standard form. . Coefficient controls opening (up if , down if ) and steepness. is the -intercept.
Factored form. . Roots are at and . Useful when roots are known.
Turning-point (vertex) form. . Vertex at . Axis of symmetry . Useful when the vertex is known.
Convert between forms by expanding (factored or vertex to standard) or completing the square (standard to vertex).
Vertex from standard form
The discriminant
Classifies the roots of :
| IMATH_16 | Roots | Parabola |
|---|---|---|
| IMATH_17 | Two real distinct | Crosses -axis twice |
| IMATH_19 | One real repeated | Touches -axis (vertex on it) |
| IMATH_21 | No real roots | Does not touch -axis |
Quadratic formula
Worked example
Sketch .
Vertex form. Vertex at . , so opens downward.
-intercept: .
-intercepts: or .
Common traps
Sign of in . A common slip is to forget the minus sign.
Forgetting that vertex form's has opposite sign to the bracket. has , not .
Mixing up with the leading coefficient. In , is the same coefficient as in standard form .
Treating no-real-roots as no solution. The quadratic has complex roots; they just do not appear on the real-number axis.
In one sentence
Quadratics have vertex at , classify their roots through the discriminant ( two real, one repeated, none), and can be written in standard, factored () or vertex () form depending on which features are known.
Past exam questions, worked
Real questions from past VCAA papers on this dot point, with our answer explainer.
Year 11 SAC5 marksFor $f(x) = 2x^2 - 8x + 5$, find (a) the vertex, (b) the discriminant, (c) the exact $x$-intercepts.Show worked answer β
(a) Vertex. . .
Vertex: .
(b) Discriminant. .
Two real distinct roots.
(c) -intercepts. .
Markers reward the vertex formula, discriminant evaluation, and simplification of the quadratic-formula surd.
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