Mathematics » Functions II » Quadratic Functions

# Revision of Quadratic Functions

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 shiftFor $$q>0$$, $$f(x)$$ 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 shapeFor $$a>0$$; 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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