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.
\(a<0\)
\(a>0\)
\(q>0\)
\(q=0\)
\(q<0\)
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This article has been modified from "Quadratic Functions, by Siyavula, Mathematics Grade 11, CC BY 4.0. Download the article for free at Siyavula.
