Recall that a function describes a specific relationship between two variables; where an independent (input) variable has exactly one dependent (output) variable. Every element in the domain maps to only one element in the range. Functions can be one-to-one relations or many-to-one relations. A many-to-one relation associates two or more values of the independent variable with a single value of the dependent variable. Functions allow us to visualise relationships in the form of graphs, which are much easier to read and interpret than lists of numbers.

## Revision of Quadratic Functions

#### Functions of the form \(y = a x^2 + q\)Functions of the general form \(y=a{x}^{2}+q\) are called parabolic functions, where \(a\) and \(q\) are constants.

**The effects of \(a\) and \(q\) on \(f(x) = ax^2 + q\):**

**The effect of \(q\) on vertical shift**

For \(q>0\), \(f(x)\) is shifted vertically upwards by \(q\) units.

The turning point of \(f(x)\) is above the \(x\)-axis.

For \(q<0\), \(f(x)\) is shifted vertically downwards by \(q\) units.

The turning point of \(f(x)\) is below the \(x\)-axis.

\(q\) is also the \(y\)-intercept of the parabola.

**The effect of \(a\) on shape**

For \(a>0\); the graph of \(f(x)\) is a “smile” and has a minimum turning point \((0;q)\). As the value of \(a\) becomes larger, the graph becomes narrower.

As \(a\) gets closer to \(\text{0}\), \(f(x)\) becomes wider.

For \(a<0\); the graph of \(f(x)\) is a “frown” and has a maximum turning point \((0;q)\). As the value of \(a\) becomes smaller, the graph becomes narrower.

As \(a\) gets closer to \(\text{0}\), \(f(x)\) becomes wider.

\(a<0\) \(a>0\) \(q>0\) \(q=0\) \(q<0\)

**The effects of \(a\) and \(q\) on \(f(x) = ax^2 + q\):**

**The effect of \(q\) on vertical shift**

For \(q>0\), \(f(x)\) is shifted vertically upwards by \(q\) units.

The turning point of \(f(x)\) is above the \(x\)-axis.

For \(q<0\), \(f(x)\) is shifted vertically downwards by \(q\) units.

The turning point of \(f(x)\) is below the \(x\)-axis.

\(q\) is also the \(y\)-intercept of the parabola.

**The effect of \(a\) on shape**

For \(a>0\); the graph of \(f(x)\) is a “smile” and has a minimum turning point \((0;q)\). As the value of \(a\) becomes larger, the graph becomes narrower.

As \(a\) gets closer to \(\text{0}\), \(f(x)\) becomes wider.

For \(a<0\); the graph of \(f(x)\) is a “frown” and has a maximum turning point \((0;q)\). As the value of \(a\) becomes smaller, the graph becomes narrower.

As \(a\) gets closer to \(\text{0}\), \(f(x)\) becomes wider.

\(a<0\) | \(a>0\) | |

\(q>0\) | ||

\(q=0\) | ||

\(q<0\) |