Mastering the Quadratic Function (OLevel's Pure Math)

1. The Basics: Standard Form

A quadratic function is a polynomial of degree 2, generally expressed as:

$$f\left(x\right)=a{x}^2+bx+c$$

Where $a$, $b$, and $c$ are constants and $a\ne 0$.

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2. Features of the Graph

The graph of a quadratic function is a parabola. Its orientation depends on the coefficient $a$:

• If $a>0$, the parabola opens upwards (minimum point).
• If $a<0$, the parabola opens downwards (maximum point).

Key features include the y-intercept at $(0,c)$ and the axis of symmetry at:

$$x=-\frac{b}{2a}$$

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3. The Discriminant and the Nature of Roots

To find where the graph crosses the x-axis, we solve $a{x}^2+bx+c=0$ using the Quadratic Formula:

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

The expression under the square root, $b^2-4ac$, is called the Discriminant ($\Delta$).

Discriminant Value	Nature of Roots	Graph Interaction
$b^2-4ac>0$	Two distinct real roots	Crosses x-axis twice
$b^2-4ac=0$	One repeated real root	Touches x-axis once (tangent)
$b^2-4ac<0$	No real roots	Does not touch the x-axis

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4. Completing the Square (Vertex Form)

By completing the square, we can rewrite the function in vertex form:

$$f\left(x\right)=a(x-h{)}^2+k$$

The point $(h,k)$ is the vertex (turning point) of the parabola.

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5. Worked Example: Finding the Vertex

Question: Find the coordinates of the turning point for $f\left(x\right)=2x^2-12x+23$.

Solution:

1. Factor out the coefficient of $x^2$ from the first two terms:
   $$f\left(x\right)=2(x^2-6x)+23$$
2. Complete the square inside the bracket:
   Take half of $-6$ (which is $-3$) and square it ($9$).
   $$f\left(x\right)=2\left[\right(x-3{)}^2-9]+23$$
3. Expand and simplify:
   $$f\left(x\right)=2(x-3{)}^2-18+23$$ $$f\left(x\right)=2(x-3{)}^2+5$$
4. Identify the Vertex:
   The vertex is at $(3,5)$. Since $a=2>0$, this is a minimum point.

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Summary Checklist

• Can you identify $a$, $b$, and $c$?
• Do you know if the parabola opens up or down?
• Can you calculate the discriminant to find the number of roots?
• Can you complete the square to find the turning point?