Discovering the Characteristics of Quadratic Functions
Discovering the Characteristics of Quadratic Functions
The standard form of the equation of a parabola is \(y=a{x}^{2}+q\).
Domain and range
The domain is \(\{x:x\in \mathbb{R}\}\) because there is no value for which \(f(x)\) is undefined.
If \(a>0\) then we have:
\[\begin{array}{cccl} {x}^{2}& \ge & 0 & (\text{Perfect square is always positive}) \\ a{x}^{2}& \ge & 0 & (\text{since } a>0) \\ a{x}^{2}+q & \ge & q & (\text{add } q \text{ to both sides}) \\ \therefore f(x) & \ge & q & \end{array}\]Therefore if \(a>0\), the range is \([q;\infty )\). Similarly, if \(a<0\) then the range is \((-\infty ;q]\).
Example
Question
If \(g(x)={x}^{2}+2\), determine the domain and range of the function.
Determine the domain
The domain is \(\{x:x\in \mathbb{R}\}\) because there is no value for which \(g(x)\) is undefined.
Determine the range
The range of \(g(x)\) can be calculated as follows:
\begin{align*} {x}^{2}& \ge 0 \\ {x}^{2}+2& \ge 2\\ g(x)& \ge 2 \end{align*}Therefore the range is \(\{g(x):g(x)\ge 2\}\).
Intercepts
The \(y\)-intercept:
Every point on the \(y\)-axis has an \(x\)-coordinate of \(\text{0}\), therefore to calculate the \(y\)-intercept let \(x=0\).
For example, the \(y\)-intercept of \(g(x)={x}^{2}+2\) is given by setting \(x=0\):
\begin{align*} g(x)& = {x}^{2}+2 \\ g(0)& ={0}^{2}+2\\ & =2 \end{align*}This gives the point \((0;2)\).
The \(x\)-intercepts:
Every point on the \(x\)-axis has a \(y\)-coordinate of \(\text{0}\), therefore to calculate the \(x\)-intercept let \(y=0\).
For example, the \(x\)-intercepts of \(g(x)={x}^{2}+2\) are given by setting \(y=0\):
\begin{align*} g(x)& = {x}^{2}+2 \\ 0 & = {x}^{2}+2 \\ -2& = {x}^{2} \end{align*}There is no real solution, therefore the graph of \(g(x)={x}^{2}+2\) does not have \(x\)-intercepts.
Turning points
The turning point of the function of the form \(f(x)=a{x}^{2}+q\) is determined by examining the range of the function.
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If \(a>0\), the graph of \(f(x)\) is a “smile” and has a minimum turning point at \((0;q)\).
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If \(a<0\), the graph of \(f(x)\) is a “frown” and has a maximum turning point at \((0;q)\).
Axes of symmetry
The axis of symmetry for functions of the form \(f(x)=a{x}^{2}+q\) is the \(y\)-axis, which is the line \(x=0\).
This lesson is part of:
Functions I