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)