2.2.1 Quadratic Functions

Quadratic Functions & Graphs

A quadratic graph can be written in the form $y = a x^{2} + b x + c$ where $a \neq 0$
The value of a affects the shape of the curve
- If a is positive the shape is concave up ∪
- If a is negative the shape is concave down ∩
The y-intercept is at the point (0, c)
The zeros or roots are the solutions to $a x^{2} + b x + c = 0$
- These can be found by
  - Factorising
  - Quadratic formula
  - Using your GDC
- These are also called the x-intercepts
- There can be 0, 1 or 2 x-intercepts
  - This is determined by the value of the discriminant
There is an axis of symmetry at $x = - \frac{b}{2 a}$
- This is given in your formula booklet
- If there are two x-intercepts then the axis of symmetry goes through the midpoint of them
The vertex lies on the axis of symmetry
- It can be found by completing the square
- The x-coordinate is $x = - \frac{b}{2 a}$
- The y-coordinate can be found using the GDC or by calculating y when $x = - \frac{b}{2 a}$
- If a is positive then the vertex is the minimum point
- If a is negative then the vertex is the maximum point

$f (x) = a x^{2} + b x + c$
- This is the general form
- It clearly shows the y-intercept (0, c)
- You can find the axis of symmetry by $x = - \frac{b}{2 a}$
  - This is given in the formula booklet
$f (x) = a (x - p) (x - q)$
- This is the factorised form
- It clearly shows the roots (p, 0) & (q, 0)
- You can find the axis of symmetry by $x = \frac{p + q}{2}$
$f (x) = a {(x - h)}^{2} + k$
- This is the vertex form
- It clearly shows the vertex (h, k)
- The axis of symmetry is therefore $x = h$
- It clearly shows how the function can be transformed from the graph $y = x^{2}$
  - Vertical stretch by scale factor a
  - Translation by vector $(\begin{matrix} h \\ k \end{matrix})$

If you have the roots x = p and x = q...
- Write in factorised form $y = a (x - p) (x - q)$
- You will need a third point to find the value of a
If you have the vertex (h, k) then...
- Write in vertex form $y = a {(x - h)}^{2} + k$
- You will need a second point to find the value of a
If you have three random points (x₁, y₁), (x₂, y₂) & (x₃, y₃) then...
- Write in the general form $y = a x^{2} + b x + c$
- Substitute the three points into the equation
- Form and solve a system of three linear equations to find the values of a, b & c

Use your GDC to find the roots and the turning point of a quadratic function
- You do not need to factorise or complete the square
- It is good exam technique to sketch the graph from your GDC as part of your working

The diagram below shows the graph of $y = f (x)$ , where $f (x)$ is a quadratic function.

The intercept with the $y$ -axis and the vertex have been labelled.

2-2-1-ib-aa-sl-we-image

Write down an expression for $y = f (x)$ .

2-2-1-ib-aa-sl-quad-function-we-solution