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Quadratic Functions & Parabolas – Comprehensive Study Notes

Quadratic Functions: Core Definition

  • A quadratic function is any function that can be written in the general (a.k.a. canonical / polynomial) form f(x)=ax^2+bx+c \qquad (a\neq 0)
  • The presence of the x^2-term (with non–zero coefficient a) is essential; without it the function is no longer quadratic.
  • Basic algebraic manipulations (factoring, completing the square, etc.) always presume this starting form.

Vertex ("Standard") Form & How to Obtain It

  • By completing the square we can rewrite any quadratic in the so-called vertex / standard form: f(x)=a\bigl(x-h\bigr)^2+k

  • h and k have geometric meaning (coordinates of the vertex).

  • Formulas to compute them directly from the coefficients a,b,c (no need to re-complete the square each time):

    h=-\frac b{2a}

    k=f(h) (i.e. substitute x=h back into the original quadratic).

  • Summary of why the vertex form is useful:

  • Makes the vertex explicit (point (h,k)).

  • Makes it easy to read whether the parabola opens up or down (sign of a).

  • Simplifies graphing, optimisation, and modelling questions.

Parabola Geometry & Vocabulary

  • The graph of any quadratic function is called a parabola.

  • Exactly two possible shapes:

  • a>0 ⇒ opens up (shaped like the letter “U”); has a minimum value.

  • a<0 ⇒ opens down (shaped like an upside-down “U”); has a maximum value.

  • Key features and their formulas:

  • Vertex: (h,k) – the point where the minimum or maximum occurs.

  • Axis of symmetry (a.k.a. axis of the parabola): vertical line x=h; every parabola is symmetric about this line.

  • Intercepts
    • y-intercept: set x=0 ⇒ f(0)=c (straight from general form).
    • x-intercepts (roots): solve f(x)=0; may be 0, 1, or 2 real solutions.

  • Minimum/Maximum value: always equal to k (because it occurs at the vertex).

Quick Graphing Checklist (Exam Strategy)

  • 1. Identify a,b,c from ax^2+bx+c.

  • 2. Compute h=-\dfrac{b}{2a} and k=f(h).
    ⇒ gives vertex + min/max.

  • 3. Decide shape:

  • a>0 → opens up (minimum).

  • a<0 → opens down (maximum).

  • 4. Axis of symmetry is x=h (draw a dashed vertical line for reference).

  • 5. Intercepts:

  • y-intercept at (0,c).

  • x-intercepts by factoring, quadratic formula, or completing the square on f(x)=0.

  • 6. Plot vertex, intercepts, draw a smooth symmetric curve.

Worked Example 1 (Full Walk-Through)

Given f(x)=2x^2+8x-10

  • Step 1 – coefficients: a=2,\;b=8,\;c=-10.
  • Step 2 – vertex coordinates
    h=-\frac b{2a}=-\frac{8}{2\cdot2}=-2
    k=f(-2)=2(-2)^2+8(-2)-10=8-16-10=-18
    ⇒ vertex (-2,-18).
  • Step 3 – standard form
    f(x)=2(x+2)^2-18 (obtained either from formulas or by completing the square).
  • Step 4 – axis of symmetry
    x=-2.
  • Step 5 – intercepts
    • y-intercept: f(0)=-10 ⇒ point (0,-10).
    • x-intercepts: solve 2x^2+8x-10=0 \;\Longrightarrow\; x^2+4x-5=0 \;\Longrightarrow\; (x+5)(x-1)=0
    ⇒ x=-5 and x=1.
  • Step 6 – shape: a=2>0, parabola opens up; minimum value k=-18.
  • Sketch: vertex at (-2,-18), axis x=-2, points at (-5,0), (1,0), (0,-10); draw symmetric “U”.

Worked Example 2 (Short Form)

Function: f(x)=x^2-4x+3

  • a=1\;(&>0) → opens up → has a minimum.
  • h=-\frac{-4}{2\cdot1}=2; k=f(2)=2^2-4\cdot2+3=4-8+3=-1.
  • Axis: x=2. Vertex: (2,-1). Minimum value: -1.

Exam-Style Mini-Problems & Solutions

  1. Find a if only given the point it must pass through.
  • If the quadratic has the shape f(x)=ax^2 and contains (2,-8):
    -8=a\cdot2^2 \;\Longrightarrow\; a=-2 ⇒ f(x)=-2x^2.
  1. Vertex-plus-point reconstruction.
  • Vertex (2,0) means h=2,\;k=0.
  • Generic form: f(x)=a(x-2)^2+0.
  • Contains (1,3)
    3=a(1-2)^2=a(1)\;\Rightarrow\;a=3
    ⇒ f(x)=3(x-2)^2.
  1. Typical “axis/min or max” prompt.
  • For any f(x)=ax^2+bx+c simply cite:
  • Axis x=h=-\dfrac{b}{2a}.
  • Min/Max = k=f(h) and its nature (min vs max) decided by sign of a.

Completing the Square (Recap Procedure)

  • Starting with ax^2+bx+c:
  1. Factor out a from the first two terms:
    a\bigl(x^2+\frac bx x\bigr)+c.
  2. Inside parentheses create a perfect square:
    x^2+\frac bx x+\bigl(\frac{b}{2a}\bigr)^2, then immediately subtract the same square so the expression is unchanged.
  3. Re-distribute a to the subtracted square, tidy constants, and you arrive at a(x-h)^2+k.
  • Although the algebra is straightforward, use the formulas for h,k on timed tests – faster and mistake-proof.

Practical/Instructional Remarks

  • In class & on exams students are expected to:
  • Convert freely between general form and vertex form.
  • Locate vertex, axis, intercepts swiftly – especially under time pressure.
  • Recognise the symmetry property and use it to check algebraic work (points should appear in symmetric pairs about x=h).
  • When sketching, marking the axis (even lightly) provides an immediate visual cue for correctly shaping the curve.
  • Vertical and horizontal line notation (e.g. x=-2 vs y=5)